Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Lecture 7: Digital Signals Signals and Spectral Methods in Geoinformatics

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digital Signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digitalization of signals Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digitalization of signals Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation A1. Sampling Α2. Quantization Α3. Codification

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digitalization of signals Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation Sampling theorem A1. Sampling Α2. Quantization Α3. Codification

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digitalization of signals Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation Sampling theorem A1. Sampling Α2. Quantization Α3. Codification If m(t) is a band-limited signal ( M(ω) = 0 for |ω| > ω Μ ) then the signal m(t) can be reconstructed from sampling values (at equal distances ΔT )

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digitalization of signals Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation Sampling theorem A1. Sampling Α2. Quantization Α3. Codification provided that the sampling is dense enough, specifically when If m(t) is a band-limited signal ( M(ω) = 0 for |ω| > ω Μ ) then the signal m(t) can be reconstructed from sampling values (at equal distances ΔT )

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digitalization of signals Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation Sampling theorem A1. Sampling Α2. Quantization Α3. Codification provided that the sampling is dense enough, specifically when The signal is reconstructed through the relation If m(t) is a band-limited signal ( M(ω) = 0 for |ω| > ω Μ ) then the signal m(t) can be reconstructed from sampling values (at equal distances ΔT )

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics m(t)m(t) t Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics m(t)m(t) t Sampling determination of values m n = m(n ΔΤ) at intervals of ΔΤ Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics m(t)m(t) t ΔTΔT m1m1 m2m2 m3m3 m4m4 m5m5 Sampling determination of values m n = m(n ΔΤ) at intervals of ΔΤ Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics m(t)m(t) t ΔTΔT m1m1 m2m2 m3m3 m4m4 m5m Sampling determination of values m n = m(n ΔΤ) at intervals of ΔΤ initial value x k Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics m(t)m(t) t ΔTΔT m1m1 m2m2 m3m3 m4m4 m5m Sampling determination of values m n = m(n ΔΤ) at intervals of ΔΤ Quantization replacement of each value m n = m(n ΔΤ) with the closest value x k from a predefined discrete set..., x i, x i+1,..., x i+n,... (usually equidistant) initial value x k Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics m(t)m(t) t ΔTΔT m1m1 m2m2 m3m3 m4m4 m5m Sampling determination of values m n = m(n ΔΤ) at intervals of ΔΤ Quantization replacement of each value m n = m(n ΔΤ) with the closest value x k from a predefined discrete set..., x i, x i+1,..., x i+n,... (usually equidistant) initial value x k Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics m(t)m(t) t ΔTΔT m1m1 m2m2 m3m3 m4m4 m5m Sampling determination of values m n = m(n ΔΤ) at intervals of ΔΤ Quantization replacement of each value m n = m(n ΔΤ) with the closest value x k from a predefined discrete set..., x i, x i+1,..., x i+n,... (usually equidistant) initial value x k Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics m(t)m(t) t ΔTΔT m1m1 m2m2 m3m3 m4m4 m5m Sampling determination of values m n = m(n ΔΤ) at intervals of ΔΤ Quantization replacement of each value m n = m(n ΔΤ) with the closest value x k from a predefined discrete set..., x i, x i+1,..., x i+n,... (usually equidistant) initial value x k quantum value Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics m(t)m(t) t ΔTΔT m1m1 m2m2 m3m3 m4m4 m5m Sampling determination of values m n = m(n ΔΤ) at intervals of ΔΤ Quantization replacement of each value m n = m(n ΔΤ) with the closest value x k from a predefined discrete set..., x i, x i+1,..., x i+n,... (usually equidistant) initial value x k quantum value Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics m(t)m(t) t ΔTΔT m1m1 m2m2 m3m3 m4m4 m5m Sampling determination of values m n = m(n ΔΤ) at intervals of ΔΤ Quantization replacement of each value m n = m(n ΔΤ) with the closest value x k from a predefined discrete set..., x i, x i+1,..., x i+n,... (usually equidistant) initial value x k quantum value Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics m(t)m(t) t ΔTΔT m1m1 m2m2 m3m3 m4m4 m5m Sampling determination of values m n = m(n ΔΤ) at intervals of ΔΤ Quantization replacement of each value m n = m(n ΔΤ) with the closest value x k from a predefined discrete set..., x i, x i+1,..., x i+n,... (usually equidistant) Codification replacement of the value x k with a code, i.e. an integer k expressed in the binary system (only digits 0 and 1) initial value x k quantum value Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics m(t)m(t) t ΔTΔT m1m1 m2m2 m3m3 m4m4 m5m Sampling determination of values m n = m(n ΔΤ) at intervals of ΔΤ Quantization replacement of each value m n = m(n ΔΤ) with the closest value x k from a predefined discrete set..., x i, x i+1,..., x i+n,... (usually equidistant) Codification replacement of the value x k with a code, i.e. an integer k expressed in the binary system (only digits 0 and 1) initial value x k quantum value code Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics m(t)m(t) t ΔTΔT m1m1 m2m2 m3m3 m4m4 m5m Sampling determination of values m n = m(n ΔΤ) at intervals of ΔΤ Quantization replacement of each value m n = m(n ΔΤ) with the closest value x k from a predefined discrete set..., x i, x i+1,..., x i+n,... (usually equidistant) Codification replacement of the value x k with a code, i.e. an integer k expressed in the binary system (only digits 0 and 1) initial value x k quantum value code Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics m(t)m(t) t ΔTΔT m1m1 m2m2 m3m3 m4m4 m5m Sampling determination of values m n = m(n ΔΤ) at intervals of ΔΤ Quantization replacement of each value m n = m(n ΔΤ) with the closest value x k from a predefined discrete set..., x i, x i+1,..., x i+n,... (usually equidistant) Codification replacement of the value x k with a code, i.e. an integer k expressed in the binary system (only digits 0 and 1) initial value x k binary code quantum value code Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Sampling determination of values m n = m(n ΔΤ) at intervals of ΔΤ m(t)m(t) t Quantization replacement of each value m n = m(n ΔΤ) with the closest value x k from a predefined discrete set..., x i, x i+1,..., x i+n,... (usually equidistant) Codification replacement of the value x k with a code, i.e. an integer k expressed in the binary system (only digits 0 and 1) ΔTΔT m1m1 m2m2 m3m3 m4m4 m5m initial value x k binary code quantum value code Digitalization of signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Signaling Format

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Transmission of digital signals Signaling Format

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Transmission of digital signals Binary signal to be transmitted = sequence {b i } with b i = 0 or b i =1 Signaling Format

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Transmission of digital signals Binary signal to be transmitted = sequence {b i } with b i = 0 or b i =1 Transmission with new signal m(t) with possible values  1, 0, 1 Signaling Format

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Transmission of digital signals Binary signal to be transmitted = sequence {b i } with b i = 0 or b i =1 Transmission with new signal m(t) with possible values  1, 0, 1 A time interval δt is assigned to every digit b i divided to 2 equal parts Signaling Format

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Transmission of digital signals Binary signal to be transmitted = sequence {b i } with b i = 0 or b i =1 Transmission with new signal m(t) with possible values  1, 0, 1 A time interval δt is assigned to every digit b i divided to 2 equal parts bibi Signaling Format

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Transmission of digital signals Binary signal to be transmitted = sequence {b i } with b i = 0 or b i =1 Transmission with new signal m(t) with possible values  1, 0, 1 A time interval δt is assigned to every digit b i divided to 2 equal parts bibi m ia m ib Signaling Format

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Transmission of digital signals Binary signal to be transmitted = sequence {b i } with b i = 0 or b i =1 Transmission with new signal m(t) with possible values  1, 0, 1 A time interval δt is assigned to every digit b i divided to 2 equal parts m(t) has values m ia and m ib (out of –1, 0, 1) in the 1st and 2nd half of the interval δt, respectively b i = 0  [m 0a, m 0b ] και b i = 1  [m 1a, m 1b ] bibi m ia m ib Signaling Format

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Signaling format = process of transforming the sequence {b i } to the sequence {m ia, m ib } The values (-1, 0 or 1) of m 0a, m 0b, m 1a, m 1b completely define the signaling format Transmission of digital signals Binary signal to be transmitted = sequence {b i } with b i = 0 or b i =1 Transmission with new signal m(t) with possible values  1, 0, 1 A time interval δt is assigned to every digit b i divided to 2 equal parts m(t) has values m ia and m ib (out of –1, 0, 1) in the 1st and 2nd half of the interval δt, respectively b i = 0  [m 0a, m 0b ] και b i = 1  [m 1a, m 1b ] bibi m ia m ib Signaling Format

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Signaling format = process of transforming the sequence {b i } to the sequence {m ia, m ib } The values (-1, 0 or 1) of m 0a, m 0b, m 1a, m 1b completely define the signaling format bibi m(t) m 0a m 0b m1am1a m1bm1b Transmission of digital signals Binary signal to be transmitted = sequence {b i } with b i = 0 or b i =1 Transmission with new signal m(t) with possible values  1, 0, 1 A time interval δt is assigned to every digit b i divided to 2 equal parts m(t) has values m ia and m ib (out of –1, 0, 1) in the 1st and 2nd half of the interval δt, respectively b i = 0  [m 0a, m 0b ] και b i = 1  [m 1a, m 1b ] bibi m ia m ib Example : Signaling Format

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Signaling format = process of transforming the sequence {b i } to the sequence {m ia, m ib } The values (-1, 0 or 1) of m 0a, m 0b, m 1a, m 1b completely define the signaling format bibi m(t) m 0a m 0b m1am1a m1bm1b Transmission of digital signals Binary signal to be transmitted = sequence {b i } with b i = 0 or b i =1 Transmission with new signal m(t) with possible values  1, 0, 1 A time interval δt is assigned to every digit b i divided to 2 equal parts m(t) has values m ia and m ib (out of –1, 0, 1) in the 1st and 2nd half of the interval δt, respectively b i = 0  [m 0a, m 0b ] και b i = 1  [m 1a, m 1b ] bibi m ia m ib Example : Signaling format: m 0a = -1, m 0b = 1, m 1a = 1, m 1b = -1 Signaling Format

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Unipolar NRZ Bipolar NRZ Unipolar RZ Bipolar RZ AMI Split-Phase (Manchester) (NRZ = Νon Return to Zero) (RZ = Return to Zero) AMI = = Alternate Mark Inversion Split-Phase (Manchester) m 0a m 0b m1am1a m1bm1b Signaling formats GPS !

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics NRZ Final transmission with one of the following 3 modulation modes

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics NRZ ASK  ASK modulation (Amplitude Shift Keying) Final transmission with one of the following 3 modulation modes

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics NRZ ASK FSK  ASK modulation (Amplitude Shift Keying)  FSK modulation (Frequency Shift Keying) Final transmission with one of the following 3 modulation modes

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics NRZ ASK FSK PSK  ASK modulation (Amplitude Shift Keying)  PSK modulation (Phase Shift Keying) GPS !  FSK modulation (Frequency Shift Keying) Final transmission with one of the following 3 modulation modes

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Spread spectrum technique

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Spread spectrum technique Modulation:Original signal d(t) with digit length T modulated as y(t) = d(t)cos( ω 0 t)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Spread spectrum technique Modulation:Original signal d(t) with digit length T modulated as y(t) = d(t)cos( ω 0 t) Coding:Multiplication with signal g(t) = ± 1 with amplitude A = 1 and digit length T g << T z(t) = g(t)d(t)cos(ω 0 t) (transmitted coded signal) Comprehensible only to those knowing the PRN code g(t)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Spread spectrum technique Modulation:Original signal d(t) with digit length T modulated as y(t) = d(t)cos( ω 0 t) Coding:Multiplication with signal g(t) = ± 1 with amplitude A = 1 and digit length T g << T z(t) = g(t)d(t)cos(ω 0 t) (transmitted coded signal) Comprehensible only to those knowing the PRN code g(t) Decoding:Multiplication of received signal z(t) with the known code g(t) g(t) z(t) = g(t) 2 d(t)cos(ω 0 t) = d(t)cos(ω 0 t) since g(t) 2 = (  1) 2 = 1 : recovery of modulated signal without the code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Spread spectrum technique Modulation:Original signal d(t) with digit length T modulated as y(t) = d(t)cos( ω 0 t) Coding:Multiplication with signal g(t) = ± 1 with amplitude A = 1 and digit length T g << T z(t) = g(t)d(t)cos(ω 0 t) (transmitted coded signal) Comprehensible only to those knowing the PRN code g(t) Decoding:Multiplication of received signal z(t) with the known code g(t) g(t) z(t) = g(t) 2 d(t)cos(ω 0 t) = d(t)cos(ω 0 t) since g(t) 2 = (  1) 2 = 1 : recovery of modulated signal without the code Demodulation: y(t) = d(t)cos(ω 0 t)  d(t) = recovery of original aignal

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Spread spectrum technique Modulation:Original signal d(t) with digit length T modulated as y(t) = d(t)cos( ω 0 t) Coding:Multiplication with signal g(t) = ± 1 with amplitude A = 1 and digit length T g << T z(t) = g(t)d(t)cos(ω 0 t) (transmitted coded signal) Comprehensible only to those knowing the PRN code g(t) Decoding:Multiplication of received signal z(t) with the known code g(t) g(t) z(t) = g(t) 2 d(t)cos(ω 0 t) = d(t)cos(ω 0 t) since g(t) 2 = (  1) 2 = 1 : recovery of modulated signal without the code Demodulation: y(t) = d(t)cos(ω 0 t)  d(t) = recovery of original aignal Bandwidth : from 2 / Τ in y(t) becomes 2 / Τ g in z(t)  T g > 2 / T

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Spread spectrum technique Modulation:Original signal d(t) with digit length T modulated as y(t) = d(t)cos( ω 0 t) Coding:Multiplication with signal g(t) = ± 1 with amplitude A = 1 and digit length T g << T z(t) = g(t)d(t)cos(ω 0 t) (transmitted coded signal) Comprehensible only to those knowing the PRN code g(t) Decoding:Multiplication of received signal z(t) with the known code g(t) g(t) z(t) = g(t) 2 d(t)cos(ω 0 t) = d(t)cos(ω 0 t) since g(t) 2 = (  1) 2 = 1 : recovery of modulated signal without the code Demodulation: y(t) = d(t)cos(ω 0 t)  d(t) = recovery of original aignal spread spectrum !Applications : Police communications, GPS Bandwidth : from 2 / Τ in y(t) becomes 2 / Τ g in z(t)  T g > 2 / T

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of digital signals

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of digital signals Digital signal = linear combination of orthogonal pulses

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of digital signals Digital signal = linear combination of orthogonal pulses Elementary orthogonal pulse (duration Τ, amplitude 1,center t = 0) :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of digital signals Digital signal = linear combination of orthogonal pulses Elementary orthogonal pulse (duration Τ, amplitude 1,center t = 0) :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of digital signals Digital signal = linear combination of orthogonal pulses Elementary orthogonal pulse (duration Τ, amplitude 1,center t = 0) :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of digital signals Digital signal = linear combination of orthogonal pulses Elementary orthogonal pulse (duration Τ, amplitude 1,center t = 0) :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of digital signals Digital signal = linear combination of orthogonal pulses Elementary orthogonal pulse (duration Τ, amplitude 1,center t = 0) :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of digital signals Digital signal = linear combination of orthogonal pulses Elementary orthogonal pulse (duration Τ, amplitude 1,center t = 0) : orthogonal pulce with center t = τ (duration Τ, amplitude 1 ) :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The signal digit containing the origin t = 0 and having center t = d (  T /2 < d < T /2) contributes to the total signal the component :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Every other digit k places after the initial (or brfore for k<0 ) has center t = d + kT, where T = digit length, has contribution : The signal digit containing the origin t = 0 and having center t = d (  T /2 < d < T /2) contributes to the total signal the component :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Every other digit k places after the initial (or brfore for k<0 ) has center t = d + kT, where T = digit length, has contribution : The signal digit containing the origin t = 0 and having center t = d (  T /2 < d < T /2) contributes to the total signal the component : Total digital signal (digits do not overlap) :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Every other digit k places after the initial (or brfore for k<0 ) has center t = d + kT, where T = digit length, has contribution : The signal digit containing the origin t = 0 and having center t = d (  T /2 < d < T /2) contributes to the total signal the component : Total digital signal (digits do not overlap) :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digital signal :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digital signal :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digital signal : Choice between the values +A and  A “random”, independently and with equal probability (= ½) : PRN = Pseudo Random NoiseGPS ! A k = random variable, x(t) = stochastic process (random function)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digital signal : Choice between the values +A and  A “random”, independently and with equal probability (= ½) : PRN = Pseudo Random NoiseGPS ! A k = random variable, x(t) = stochastic process (random function) A stochastic process x(t) taking discrete values z i (i = 1, 2,...) is characterized by the joined probabilities (for every n ) :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digital signal : Choice between the values +A and  A “random”, independently and with equal probability (= ½) : PRN = Pseudo Random NoiseGPS ! A k = random variable, x(t) = stochastic process (random function) A stochastic process x(t) taking discrete values z i (i = 1, 2,...) is characterized by the joined probabilities (for every n ) : mean function :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digital signal : Choice between the values +A and  A “random”, independently and with equal probability (= ½) : PRN = Pseudo Random NoiseGPS ! A k = random variable, x(t) = stochastic process (random function) A stochastic process x(t) taking discrete values z i (i = 1, 2,...) is characterized by the joined probabilities (for every n ) : correlation function : mean function :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Digital signal : Choice between the values +A and  A “random”, independently and with equal probability (= ½) : PRN = Pseudo Random NoiseGPS ! A k = random variable, x(t) = stochastic process (random function) A stochastic process x(t) taking discrete values z i (i = 1, 2,...) is characterized by the joined probabilities (for every n ) : covariance function : correlation function : mean function :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Stochastic characteristics of PRN noise (Pseudo Random Noise)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Stochastic characteristics of PRN noise (Pseudo Random Noise) Digital signal (PRN code) as a stochastic process (random function)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Stochastic characteristics of PRN noise (Pseudo Random Noise) Digital signal (PRN code) as a stochastic process (random function) A k = random variables with possible values + A και  A, with equal probability and independent

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Stochastic characteristics of PRN noise (Pseudo Random Noise) Digital signal (PRN code) as a stochastic process (random function) Probabilities :Probability ( A k = +A ) = 1/2 Probability ( Α k =  A ) = 1/2 A k = random variables with possible values + A και  A, with equal probability and independent

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Stochastic characteristics of PRN noise (Pseudo Random Noise) Digital signal (PRN code) as a stochastic process (random function) Joint probabilities : Probability ( A k = +A AND A j = +A ) = ½ ½ = 1/4 Probability ( Α k = +A AND A j =  A ) = ½ ½ = 1/4 Probability ( Α k =  A AND A j = +A ) = ½ ½ = 1/4 Probability ( Α k =  A AND A j =  A ) = ½ ½ = 1/4 Probabilities :Probability ( A k = +A ) = 1/2 Probability ( Α k =  A ) = 1/2 A k = random variables with possible values + A και  A, with equal probability and independent

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Stochastic characteristics of PRN noise (Pseudo Random Noise) Digital signal (PRN code) as a stochastic process (random function) mean value: m A k  E{A k } = 0 Joint probabilities : Probability ( A k = +A AND A j = +A ) = ½ ½ = 1/4 Probability ( Α k = +A AND A j =  A ) = ½ ½ = 1/4 Probability ( Α k =  A AND A j = +A ) = ½ ½ = 1/4 Probability ( Α k =  A AND A j =  A ) = ½ ½ = 1/4 Probabilities :Probability ( A k = +A ) = 1/2 Probability ( Α k =  A ) = 1/2 A k = random variables with possible values + A και  A, with equal probability and independent

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Stochastic characteristics of PRN noise (Pseudo Random Noise) Digital signal (PRN code) as a stochastic process (random function) variance: σ A k 2  E{(A k  m A k ) 2 } = E{A k 2 } = A 2 mean value: m A k  E{A k } = 0 Joint probabilities : Probability ( A k = +A AND A j = +A ) = ½ ½ = 1/4 Probability ( Α k = +A AND A j =  A ) = ½ ½ = 1/4 Probability ( Α k =  A AND A j = +A ) = ½ ½ = 1/4 Probability ( Α k =  A AND A j =  A ) = ½ ½ = 1/4 Probabilities :Probability ( A k = +A ) = 1/2 Probability ( Α k =  A ) = 1/2 A k = random variables with possible values + A και  A, with equal probability and independent

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Stochastic characteristics of PRN noise (Pseudo Random Noise) Digital signal (PRN code) as a stochastic process (random function) covariance: σ A k A j  E{(A k  m A k )(A j  m A j )} = E{A k A j } = 0(k  j) variance: σ A k 2  E{(A k  m A k ) 2 } = E{A k 2 } = A 2 mean value: m A k  E{A k } = 0 Joint probabilities : Probability ( A k = +A AND A j = +A ) = ½ ½ = 1/4 Probability ( Α k = +A AND A j =  A ) = ½ ½ = 1/4 Probability ( Α k =  A AND A j = +A ) = ½ ½ = 1/4 Probability ( Α k =  A AND A j =  A ) = ½ ½ = 1/4 Probabilities :Probability ( A k = +A ) = 1/2 Probability ( Α k =  A ) = 1/2 A k = random variables with possible values + A και  A, with equal probability and independent

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Determination of the stochastic characteristics of PRN code Digital signal (PRN code) as a stochastic process (random function)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Determination of the stochastic characteristics of PRN code Digital signal (PRN code) as a stochastic process (random function) A k = randomvariables with possible values x = ± A, with equal probability and independent:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Probabilities : Determination of the stochastic characteristics of PRN code Digital signal (PRN code) as a stochastic process (random function) A k = randomvariables with possible values x = ± A, with equal probability and independent:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Probabilities : Determination of the stochastic characteristics of PRN code Digital signal (PRN code) as a stochastic process (random function) A k = randomvariables with possible values x = ± A, with equal probability and independent: Joint probabilities ( x = ± A, y = ± A ) :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics mean value: Probabilities : Determination of the stochastic characteristics of PRN code Digital signal (PRN code) as a stochastic process (random function) A k = randomvariables with possible values x = ± A, with equal probability and independent: Joint probabilities ( x = ± A, y = ± A ) :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics mean value: Probabilities : Determination of the stochastic characteristics of PRN code Digital signal (PRN code) as a stochastic process (random function) A k = randomvariables with possible values x = ± A, with equal probability and independent: Joint probabilities ( x = ± A, y = ± A ) : variance:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics mean value: Probabilities : Determination of the stochastic characteristics of PRN code Digital signal (PRN code) as a stochastic process (random function) A k = randomvariables with possible values x = ± A, with equal probability and independent: Joint probabilities ( x = ± A, y = ± A ) : variance: covariance:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics mean value: Probabilities : Determination of the stochastic characteristics of PRN code Digital signal (PRN code) as a stochastic process (random function) A k = randomvariables with possible values x = ± A, with equal probability and independent: Joint probabilities ( x = ± A, y = ± A ) : variance: covariance:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics mean value: Probabilities : Determination of the stochastic characteristics of PRN code Digital signal (PRN code) as a stochastic process (random function) A k = randomvariables with possible values x = ± A, with equal probability and independent: Joint probabilities ( x = ± A, y = ± A ) : variance: covariance:

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics where : Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The stochastic process x(t) has mean value where : Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The stochastic process x(t) has mean value because where : Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The stochastic process x(t) has mean value because and covariance function where : Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The stochastic process x(t) has mean value because and covariance function where : Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The stochastic process x(t) has mean value because and covariance function where : Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The stochastic process x(t) has mean value because and covariance function where : Therefore : when t 1 and t 2 belong to the same digit k Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The stochastic process x(t) has mean value because and covariance function where : Therefore : when t 1 and t 2 belong to the same digit k when t 1 and t 2 belong to different digits k 1  k 2 Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics The stochastic process x(t) has mean value because and covariance function To determine whether t 1 and t 2 are within the same digit since R(t 1, t 2 ) = R(t 2, t 1 ), we shall examine only the case t 1 < t 2 (otherwise we swap t 1 and t 2 ) where : Therefore : when t 1 and t 2 belong to the same digit k when t 1 and t 2 belong to different digits k 1  k 2 Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics There exist 3 cases Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics There exist 3 cases (Α) τ = t 2 – t 1 > T t 1, t 2 always belong to different digits TτTτ τ τδ t1t1 t2t2 Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics There exist 3 cases (Α) τ = t 2 – t 1 > T t 1, t 2 always belong to different digits TτTτ τ τδ t1t1 t2t2 (Β) τ = t 2 – t 1 < T t 1, t 2 belong to different digits TτTτ τ τδ t1t1 t2t2 Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics There exist 3 cases (Α) τ = t 2 – t 1 > T t 1, t 2 always belong to different digits (C) τ = t 2 – t 1 < T t 1, t 2 belong to the same digit TτTτ τ τδ t1t1 t2t2 TτTτ τ τδ t1t1 t2t2 (Β) τ = t 2 – t 1 < T t 1, t 2 belong to different digits TτTτ τ τδ t1t1 t2t2 Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics There exist 3 cases (Α) τ = t 2 – t 1 > T t 1, t 2 always belong to different digits (C) τ = t 2 – t 1 < T t 1, t 2 belong to the same digit TτTτ τ τδ t1t1 t2t2 TτTτ τ τδ t1t1 t2t2 (Β) τ = t 2 – t 1 < T t 1, t 2 belong to different digits TτTτ τ τδ t1t1 t2t2 δ + τ > Τ  Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics There exist 3 cases (Α) τ = t 2 – t 1 > T t 1, t 2 always belong to different digits (C) τ = t 2 – t 1 < T t 1, t 2 belong to the same digit TτTτ τ τδ t1t1 t2t2 TτTτ τ τδ t1t1 t2t2 (Β) τ = t 2 – t 1 < T t 1, t 2 belong to different digits δ > T  τ TτTτ τ τδ t1t1 t2t2 δ + τ > Τ  Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics There exist 3 cases (Α) τ = t 2 – t 1 > T t 1, t 2 always belong to different digits (C) τ = t 2 – t 1 < T t 1, t 2 belong to the same digit TτTτ τ τδ t1t1 t2t2 TτTτ τ τδ t1t1 t2t2 (Β) τ = t 2 – t 1 < T t 1, t 2 belong to different digits δ > T  τ TτTτ τ τδ t1t1 t2t2 δ + τ > Τ  δ + τ < Τ  Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics There exist 3 cases (Α) τ = t 2 – t 1 > T t 1, t 2 always belong to different digits (C) τ = t 2 – t 1 < T t 1, t 2 belong to the same digit δ < T  τ TτTτ τ τδ t1t1 t2t2 TτTτ τ τδ t1t1 t2t2 (Β) τ = t 2 – t 1 < T t 1, t 2 belong to different digits δ > T  τ TτTτ τ τδ t1t1 t2t2 δ + τ > Τ  δ + τ < Τ  Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics There exist 3 cases (Α) τ = t 2 – t 1 > T t 1, t 2 always belong to different digits (C) τ = t 2 – t 1 < T t 1, t 2 belong to the same digit δ < T  τ TτTτ τ τδ t1t1 t2t2 TτTτ τ τδ t1t1 t2t2 (Β) τ = t 2 – t 1 < T t 1, t 2 belong to different digits δ > T  τ TτTτ τ τδ t1t1 t2t2 t 1, t 2 in different digits, the random variables and are independent and δ + τ > Τ  δ + τ < Τ  Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics There exist 3 cases (Α) τ = t 2 – t 1 > T t 1, t 2 always belong to different digits (C) τ = t 2 – t 1 < T t 1, t 2 belong to the same digit δ < T  τ TτTτ τ τδ t1t1 t2t2 TτTτ τ τδ t1t1 t2t2 (Β) τ = t 2 – t 1 < T t 1, t 2 belong to different digits δ > T  τ TτTτ τ τδ t1t1 t2t2 t 1, t 2 in different digits, the random variables and are independent and t 1, t 2 in neighboring digits, the random variables and are independent and δ + τ > Τ  δ + τ < Τ  Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics There exist 3 cases (Α) τ = t 2 – t 1 > T t 1, t 2 always belong to different digits (C) τ = t 2 – t 1 < T t 1, t 2 belong to the same digit δ < T  τ TτTτ τ τδ t1t1 t2t2 TτTτ τ τδ t1t1 t2t2 (Β) τ = t 2 – t 1 < T t 1, t 2 belong to different digits δ > T  τ TτTτ τ τδ t1t1 t2t2 t 1, t 2 in the same digit, the random variables and are identical t 1, t 2 in different digits, the random variables and are independent and t 1, t 2 in neighboring digits, the random variables and are independent and δ + τ > Τ  δ + τ < Τ  Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics B: T  τ < δ < Τ C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics TτTτ τ τδ t1t1 t2t2 B: T  τ < δ < Τ C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics TτTτ τ τδ t1t1 t2t2 B: T  τ < δ < Τ C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics TτTτ τ τδ t1t1 t2t2 B: T  τ < δ < Τ TτTτ τ τδ t1t1 t2t2 C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics TτTτ τ τδ t1t1 t2t2 B: T  τ < δ < Τ TτTτ τ τδ t1t1 t2t2 C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics δ = random variable with homogeneous distribution in the interval ( 0 < δ < T ) : TτTτ τ τδ t1t1 t2t2 B: T  τ < δ < Τ TτTτ τ τδ t1t1 t2t2 C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics δ = random variable with homogeneous distribution in the interval ( 0 < δ < T ) : Event C = ( t 1, t 2 in the same digit) TτTτ τ τδ t1t1 t2t2 B: T  τ < δ < Τ TτTτ τ τδ t1t1 t2t2 C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics δ = random variable with homogeneous distribution in the interval ( 0 < δ < T ) : Event C = ( t 1, t 2 in the same digit) TτTτ τ τδ t1t1 t2t2 B: T  τ < δ < Τ TτTτ τ τδ t1t1 t2t2 C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics δ = random variable with homogeneous distribution in the interval ( 0 < δ < T ) : Event C = ( t 1, t 2 in the same digit)Event B = ( t 1, t 2 in different digits) TτTτ τ τδ t1t1 t2t2 B: T  τ < δ < Τ TτTτ τ τδ t1t1 t2t2 C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics δ = random variable with homogeneous distribution in the interval ( 0 < δ < T ) : Event C = ( t 1, t 2 in the same digit)Event B = ( t 1, t 2 in different digits) TτTτ τ τδ t1t1 t2t2 B: T  τ < δ < Τ TτTτ τ τδ t1t1 t2t2 C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics δ = random variable with homogeneous distribution in the interval ( 0 < δ < T ) : Event C = ( t 1, t 2 in the same digit)Event B = ( t 1, t 2 in different digits) TτTτ τ τδ t1t1 t2t2 B: T  τ < δ < Τ TτTτ τ τδ t1t1 t2t2 C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics δ = random variable with homogeneous distribution in the interval ( 0 < δ < T ) : Event C = ( t 1, t 2 in the same digit)Event B = ( t 1, t 2 in different digits) TτTτ τ τδ t1t1 t2t2 B: T  τ < δ < Τ TτTτ τ τδ t1t1 t2t2 C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics δ = random variable with homogeneous distribution in the interval ( 0 < δ < T ) : Event C = ( t 1, t 2 in the same digit)Event B = ( t 1, t 2 in different digits) TτTτ τ τδ t1t1 t2t2 B: T  τ < δ < Τ TτTτ τ τδ t1t1 t2t2 C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics δ = random variable with homogeneous distribution in the interval ( 0 < δ < T ) : Event C = ( t 1, t 2 in the same digit)Event B = ( t 1, t 2 in different digits) TτTτ τ τδ t1t1 t2t2 B: T  τ < δ < Τ TτTτ τ τδ t1t1 t2t2 C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics δ = random variable with homogeneous distribution in the interval ( 0 < δ < T ) : Event C = ( t 1, t 2 in the same digit)Event B = ( t 1, t 2 in different digits) TτTτ τ τδ t1t1 t2t2 B: T  τ < δ < Τ TτTτ τ τδ t1t1 t2t2 C: 0 < δ < T  τ Determination of the stochastic characteristics of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of PRN code Correlation of PRN code : R(τ) Α2Α2 Τ ΤΤ τ

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Corresponding spectral density : S(ω) Α2ΤΑ2Τ ω Correlation of PRN code Correlation of PRN code : R(τ) Α2Α2 Τ ΤΤ τ

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of a GPS signal with a receiver-produced copy

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of a GPS signal with a receiver-produced copy Assumption : The stochastic process x(t) is ergodic :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of a GPS signal with a receiver-produced copy Assumption : The stochastic process x(t) is ergodic :

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of a GPS signal with a receiver-produced copy Assumption : The stochastic process x(t) is ergodic : stochastic correlation same as correlation of power signal

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of a GPS signal with a receiver-produced copy Assumption : The stochastic process x(t) is ergodic : stochastic correlation same as correlation of power signal At the receiver at epoch t arrives the signal x(t – τ*) τ* = time interval related to the satellite-receiver distance ρ = c τ*, c = velocity of light (in vacuum)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of a GPS signal with a receiver-produced copy Assumption : The stochastic process x(t) is ergodic : At the receiver : [ Multiplication with signal x(t – τ) with variable delay τ ] + [integration] stochastic correlation same as correlation of power signal At the receiver at epoch t arrives the signal x(t – τ*) τ* = time interval related to the satellite-receiver distance ρ = c τ*, c = velocity of light (in vacuum)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of a GPS signal with a receiver-produced copy Assumption : The stochastic process x(t) is ergodic : At the receiver : [ Multiplication with signal x(t – τ) with variable delay τ ] + [integration] ( L = code length in digits) stochastic correlation same as correlation of power signal At the receiver at epoch t arrives the signal x(t – τ*) τ* = time interval related to the satellite-receiver distance ρ = c τ*, c = velocity of light (in vacuum)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of a GPS signal with a receiver-produced copy Assumption : The stochastic process x(t) is ergodic : At the receiver : [ Multiplication with signal x(t – τ) with variable delay τ ] + [integration] ( L = code length in digits) Variation of τ, until και stochastic correlation same as correlation of power signal At the receiver at epoch t arrives the signal x(t – τ*) τ* = time interval related to the satellite-receiver distance ρ = c τ*, c = velocity of light (in vacuum)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of a GPS signal with a receiver-produced copy Assumption : The stochastic process x(t) is ergodic : At the receiver : [ Multiplication with signal x(t – τ) with variable delay τ ] + [integration] ( L = code length in digits) Variation of τ, until και τ* has been determined and hence the (pseudo)distance ρ = c τ* ! stochastic correlation same as correlation of power signal At the receiver at epoch t arrives the signal x(t – τ*) τ* = time interval related to the satellite-receiver distance ρ = c τ*, c = velocity of light (in vacuum)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of a GPS signal with a receiver-produced copy Assumption : The stochastic process x(t) is ergodic : At the receiver : [ Multiplication with signal x(t – τ) with variable delay τ ] + [integration] ( L = code length in digits) Variation of τ, until και τ* has been determined and hence the (pseudo)distance ρ = c τ* ! Proof that stochastic correlation same as correlation of power signal At the receiver at epoch t arrives the signal x(t – τ*) τ* = time interval related to the satellite-receiver distance ρ = c τ*, c = velocity of light (in vacuum)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation of a GPS signal with a receiver-produced copy Assumption : The stochastic process x(t) is ergodic : At the receiver : [ Multiplication with signal x(t – τ) with variable delay τ ] + [integration] ( L = code length in digits) Variation of τ, until και τ* has been determined and hence the (pseudo)distance ρ = c τ* ! Proof that stochastic correlation same as correlation of power signal At the receiver at epoch t arrives the signal x(t – τ*) τ* = time interval related to the satellite-receiver distance ρ = c τ*, c = velocity of light (in vacuum)

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation function

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation function

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation function

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation function

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics + – + Correlation function

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics + – + Correlation function

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics R(τ) + + – + Correlation function

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics R(τ) + R(τ)  + – + Correlation function

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics R(τ) = R(τ) + + R(τ)  R(τ) + R(τ)  + – + Correlation function

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics Correlation function =– R(τ)R(τ)R(τ) + | R(τ) || R(τ) | R(τ) = R(τ) + + R(τ)  R(τ) + R(τ)  + – +

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics τ R(τ)R(τ) 0 TT T x(t)x(t) x(tτ)x(tτ) x(t) x(tτ)x(t) x(tτ) R(τ)+R(τ)+ R(τ)R(τ) Correlation of PRN code τ =  5Τ /4 R(τ)=R(τ) +  R(τ) 

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics τ R(τ)R(τ) 0 TT T x(t)x(t) x(tτ)x(tτ) x(t) x(tτ)x(t) x(tτ) R(τ)+R(τ)+ R(τ)R(τ) τ =  Τ R(τ)=R(τ) +  R(τ)  Correlation of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics τ R(τ)R(τ) 0 TT T x(t)x(t) x(tτ)x(tτ) x(t) x(tτ)x(t) x(tτ) R(τ)+R(τ)+ R(τ)R(τ) R(τ)=R(τ) +  R(τ)  τ =  3Τ /4 Correlation of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics τ R(τ)R(τ) 0 TT T x(t)x(t) x(tτ)x(tτ) x(t) x(tτ)x(t) x(tτ) R(τ)+R(τ)+ R(τ)R(τ) R(τ)=R(τ) +  R(τ)  τ =  2Τ /4 Correlation of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics τ R(τ)R(τ) 0 TT T x(t)x(t) x(tτ)x(tτ) x(t) x(tτ)x(t) x(tτ) R(τ)+R(τ)+ R(τ)R(τ) R(τ)=R(τ) +  R(τ)  τ =  Τ /4 Correlation of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics τ R(τ)R(τ) 0 TT T x(t)x(t) x(tτ)x(tτ) x(t) x(tτ)x(t) x(tτ) R(τ)+R(τ)+ R(τ)R(τ) R(τ)=R(τ) +  R(τ)  τ = 0 Correlation of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics τ R(τ)R(τ) 0 TT T x(t)x(t) x(tτ)x(tτ) x(t) x(tτ)x(t) x(tτ) R(τ)+R(τ)+ R(τ)R(τ) R(τ)=R(τ) +  R(τ)  τ = Τ /4 Correlation of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics τ R(τ)R(τ) 0 TT T x(t)x(t) x(tτ)x(tτ) x(t) x(tτ)x(t) x(tτ) R(τ)+R(τ)+ R(τ)R(τ) R(τ)=R(τ) +  R(τ)  τ = 2Τ /4 Correlation of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics τ R(τ)R(τ) 0 TT T x(t)x(t) x(tτ)x(tτ) x(t) x(tτ)x(t) x(tτ) R(τ)+R(τ)+ R(τ)R(τ) R(τ)=R(τ) +  R(τ)  τ = 3Τ /4 Correlation of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics τ R(τ)R(τ) 0 TT T x(t)x(t) x(tτ)x(tτ) x(t) x(tτ)x(t) x(tτ) R(τ)+R(τ)+ R(τ)R(τ) τ = Τ R(τ)=R(τ) +  R(τ)  Correlation of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics τ R(τ)R(τ) 0 TT T x(t)x(t) x(tτ)x(tτ) x(t) x(tτ)x(t) x(tτ) R(τ)+R(τ)+ R(τ)R(τ) τ = 5Τ /4 R(τ)=R(τ) +  R(τ)  Correlation of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics τ R(τ)R(τ) 0 TT T x(t)x(t) x(tτ)x(tτ) x(t) x(tτ)x(t) x(tτ) R(τ)+R(τ)+ R(τ)R(τ) τ = 5Τ /4 R(τ)=R(τ) +  R(τ)  Correlation of PRN code

Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals and Spectral Methods in Geoinformatics END