Signals and Spectral Methods

Signals and Spectral Methods
in Geoinformatics Lecture 7: Digital Signals

Digital Signals 1 1 1

Digitalization of signals

Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation

Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation A1. Sampling Α2. Quantization Α3. Codification

Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation A1. Sampling Α2. Quantization Α3. Codification Sampling theorem

Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation A1. Sampling Α2. Quantization Α3. Codification Sampling theorem If m(t) is a band-limited signal ( M(ω) = 0 for |ω| > ωΜ ) then the signal m(t) can be reconstructed from sampling values (at equal distances ΔT )

Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation A1. Sampling Α2. Quantization Α3. Codification Sampling theorem If m(t) is a band-limited signal ( M(ω) = 0 for |ω| > ωΜ ) then the signal m(t) can be reconstructed from sampling values (at equal distances ΔT ) provided that the sampling is dense enough, specifically when

Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation A1. Sampling Α2. Quantization Α3. Codification Sampling theorem If m(t) is a band-limited signal ( M(ω) = 0 for |ω| > ωΜ ) then the signal m(t) can be reconstructed from sampling values (at equal distances ΔT ) provided that the sampling is dense enough, specifically when The signal is reconstructed through the relation

m(t) t

Sampling determination of values mn = m(n ΔΤ) at intervals of ΔΤ m(t) t

Sampling determination of values mn = m(n ΔΤ) at intervals of ΔΤ m(t) t m1 m2 m3 m4 m5 ΔT

Sampling determination of values mn = m(n ΔΤ) at intervals of ΔΤ m(t) t m1 m2 m3 m4 m5 ΔT initial value xk -0.96 -2.33 -1.82 0.14 2.43

Sampling determination of values mn = m(n ΔΤ) at intervals of ΔΤ m(t) Quantization replacement of each value mn = m(n ΔΤ) with the closest value xk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) t m1 m2 m3 m4 m5 ΔT initial value xk -0.96 -2.33 -1.82 0.14 2.43

Sampling determination of values mn = m(n ΔΤ) at intervals of ΔΤ 4 m(t) 3 2 Quantization replacement of each value mn = m(n ΔΤ) with the closest value xk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 1 -1 -2 -3 t m1 m2 m3 m4 m5 ΔT initial value xk -0.96 -2.33 -1.82 0.14 2.43

Sampling determination of values mn = m(n ΔΤ) at intervals of ΔΤ 4 m(t) 3 2 Quantization replacement of each value mn = m(n ΔΤ) with the closest value xk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 1 -1 -2 -3 t m1 m2 m3 m4 m5 ΔT initial value xk -0.96 -2.33 -1.82 0.14 2.43 quantum value

Sampling determination of values mn = m(n ΔΤ) at intervals of ΔΤ 4 m(t) 3 2 Quantization replacement of each value mn = m(n ΔΤ) with the closest value xk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 1 -1 -2 -3 t m1 m2 m3 m4 m5 ΔT initial value xk -0.96 -2.33 -1.82 0.14 2.43 quantum value -1 -2 2

Sampling determination of values mn = m(n ΔΤ) at intervals of ΔΤ 4 m(t) 3 2 Quantization replacement of each value mn = m(n ΔΤ) with the closest value xk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 1 -1 -2 -3 t Codification replacement of the value xk with a code, i.e. an integer k expressed in the binary system (only digits 0 and 1) m1 m2 m3 m4 m5 ΔT initial value xk -0.96 -2.33 -1.82 0.14 2.43 quantum value -1 -2 2

Sampling determination of values mn = m(n ΔΤ) at intervals of ΔΤ 7 6 5 4 3 2 1 4 m(t) 3 2 Quantization replacement of each value mn = m(n ΔΤ) with the closest value xk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 1 -1 -2 -3 t Codification replacement of the value xk with a code, i.e. an integer k expressed in the binary system (only digits 0 and 1) m1 m2 m3 m4 m5 ΔT initial value xk -0.96 -2.33 -1.82 0.14 2.43 quantum value -1 -2 2 code

Sampling determination of values mn = m(n ΔΤ) at intervals of ΔΤ 7 6 5 4 3 2 1 4 m(t) 3 2 Quantization replacement of each value mn = m(n ΔΤ) with the closest value xk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 1 -1 -2 -3 t Codification replacement of the value xk with a code, i.e. an integer k expressed in the binary system (only digits 0 and 1) m1 m2 m3 m4 m5 ΔT initial value xk -0.96 -2.33 -1.82 0.14 2.43 quantum value -1 -2 2 code 2 1 3 5

Sampling determination of values mn = m(n ΔΤ) at intervals of ΔΤ 110 101 100 011 010 001 000 111 7 6 5 4 3 2 1 4 m(t) 3 2 Quantization replacement of each value mn = m(n ΔΤ) with the closest value xk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 1 -1 -2 -3 t Codification replacement of the value xk with a code, i.e. an integer k expressed in the binary system (only digits 0 and 1) m1 m2 m3 m4 m5 ΔT initial value xk -0.96 -2.33 -1.82 0.14 2.43 quantum value -1 -2 2 code 2 1 3 5 binary code

Sampling determination of values mn = m(n ΔΤ) at intervals of ΔΤ 110 101 100 011 010 001 000 111 7 6 5 4 3 2 1 4 m(t) 3 2 Quantization replacement of each value mn = m(n ΔΤ) with the closest value xk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 1 -1 -2 -3 t Codification replacement of the value xk with a code, i.e. an integer k expressed in the binary system (only digits 0 and 1) m1 m2 m3 m4 m5 ΔT initial value xk -0.96 -2.33 -1.82 0.14 2.43 quantum value -1 -2 2 code 2 1 3 5 binary code 010 001 011 101

Signaling Format

Signaling Format Transmission of digital signals

Signaling Format Transmission of digital signals
Binary signal to be transmitted = sequence {bi} with bi = 0 or bi =1

Binary signal to be transmitted = sequence {bi} with bi = 0 or bi =1 Transmission with new signal m(t) with possible values 1, 0, 1

Binary signal to be transmitted = sequence {bi} with bi = 0 or bi =1 Transmission with new signal m(t) with possible values 1, 0, 1 A time interval δt is assigned to every digit bi divided to 2 equal parts

Signaling Format Transmission of digital signals bi
Binary signal to be transmitted = sequence {bi} with bi = 0 or bi =1 Transmission with new signal m(t) with possible values 1, 0, 1 A time interval δt is assigned to every digit bi divided to 2 equal parts

Binary signal to be transmitted = sequence {bi} with bi = 0 or bi =1 Transmission with new signal m(t) with possible values 1, 0, 1 mia mib A time interval δt is assigned to every digit bi divided to 2 equal parts

Binary signal to be transmitted = sequence {bi} with bi = 0 or bi =1 Transmission with new signal m(t) with possible values 1, 0, 1 mia mib A time interval δt is assigned to every digit bi divided to 2 equal parts m(t) has values mia and mib (out of –1, 0, 1) in the 1st and 2nd half of the interval δt, respectively bi = 0  [m0a, m0b] και bi = 1  [m1a, m1b]

Binary signal to be transmitted = sequence {bi} with bi = 0 or bi =1 Transmission with new signal m(t) with possible values 1, 0, 1 mia mib A time interval δt is assigned to every digit bi divided to 2 equal parts m(t) has values mia and mib (out of –1, 0, 1) in the 1st and 2nd half of the interval δt, respectively bi = 0  [m0a, m0b] και bi = 1  [m1a, m1b] Signaling format = process of transforming the sequence {bi} to the sequence {mia, mib} The values (-1, 0 or 1) of m0a, m0b, m1a, m1b completely define the signaling format

Binary signal to be transmitted = sequence {bi} with bi = 0 or bi =1 Transmission with new signal m(t) with possible values 1, 0, 1 mia mib A time interval δt is assigned to every digit bi divided to 2 equal parts m(t) has values mia and mib (out of –1, 0, 1) in the 1st and 2nd half of the interval δt, respectively bi = 0  [m0a, m0b] και bi = 1  [m1a, m1b] Signaling format = process of transforming the sequence {bi} to the sequence {mia, mib} The values (-1, 0 or 1) of m0a, m0b, m1a, m1b completely define the signaling format Example : bi 1 1 1 1 m(t) m1a m1b m0a m0b

Binary signal to be transmitted = sequence {bi} with bi = 0 or bi =1 Transmission with new signal m(t) with possible values 1, 0, 1 mia mib A time interval δt is assigned to every digit bi divided to 2 equal parts m(t) has values mia and mib (out of –1, 0, 1) in the 1st and 2nd half of the interval δt, respectively bi = 0  [m0a, m0b] και bi = 1  [m1a, m1b] Signaling format = process of transforming the sequence {bi} to the sequence {mia, mib} The values (-1, 0 or 1) of m0a, m0b, m1a, m1b completely define the signaling format Example : bi 1 1 1 1 m(t) Signaling format: m0a = -1, m0b = 1, m1a = 1, m1b = -1 m1a m1b m0a m0b

Signaling formats m1a m1b m0a m0b 1 1 1 1 -1 -1 GPS ! 1 1 -1 1 -1 1 -1
1 1 1 m1a m1b m0a m0b Unipolar NRZ 1 1 (NRZ = Νon Return to Zero) Bipolar NRZ 1 1 -1 -1 GPS ! Unipolar RZ 1 (RZ = Return to Zero) 1 -1 Bipolar RZ AMI 1 AMI = = Alternate Mark Inversion -1 Split-Phase (Manchester) 1 -1 -1 1 Split-Phase (Manchester)

Final transmission with one of the following 3 modulation modes
1 1 1 NRZ

1 1 1 NRZ  ASK modulation (Amplitude Shift Keying) ASK

1 1 1 NRZ  ASK modulation (Amplitude Shift Keying) ASK  FSK modulation (Frequency Shift Keying) FSK

1 1 1 NRZ  ASK modulation (Amplitude Shift Keying) ASK  FSK modulation (Frequency Shift Keying) FSK  PSK modulation (Phase Shift Keying) GPS ! PSK

Spread spectrum technique

Modulation: Original signal d(t) with digit length T modulated as y(t) = d(t)cos(ω0t)

Modulation: Original signal d(t) with digit length T modulated as y(t) = d(t)cos(ω0t) Coding: Multiplication with signal g(t) = ± 1 with amplitude A = 1 and digit length Tg << T z(t) = g(t)d(t)cos(ω0t) (transmitted coded signal) Comprehensible only to those knowing the PRN code g(t)

Modulation: Original signal d(t) with digit length T modulated as y(t) = d(t)cos(ω0t) Coding: Multiplication with signal g(t) = ± 1 with amplitude A = 1 and digit length Tg << T z(t) = g(t)d(t)cos(ω0t) (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(ω0t) = d(t)cos(ω0t) since g(t)2 = (1)2 = 1 : recovery of modulated signal without the code

Modulation: Original signal d(t) with digit length T modulated as y(t) = d(t)cos(ω0t) Coding: Multiplication with signal g(t) = ± 1 with amplitude A = 1 and digit length Tg << T z(t) = g(t)d(t)cos(ω0t) (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(ω0t) = d(t)cos(ω0t) since g(t)2 = (1)2 = 1 : recovery of modulated signal without the code Demodulation: y(t) = d(t)cos(ω0t)  d(t) = recovery of original aignal

Modulation: Original signal d(t) with digit length T modulated as y(t) = d(t)cos(ω0t) Coding: Multiplication with signal g(t) = ± 1 with amplitude A = 1 and digit length Tg << T z(t) = g(t)d(t)cos(ω0t) (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(ω0t) = d(t)cos(ω0t) since g(t)2 = (1)2 = 1 : recovery of modulated signal without the code Demodulation: y(t) = d(t)cos(ω0t)  d(t) = recovery of original aignal Bandwidth : from 2 / Τ in y(t) becomes 2 / Τg in z(t)  Tg << T  / Tg >> 2 / T

Modulation: Original signal d(t) with digit length T modulated as y(t) = d(t)cos(ω0t) Coding: Multiplication with signal g(t) = ± 1 with amplitude A = 1 and digit length Tg << T z(t) = g(t)d(t)cos(ω0t) (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(ω0t) = d(t)cos(ω0t) since g(t)2 = (1)2 = 1 : recovery of modulated signal without the code Demodulation: y(t) = d(t)cos(ω0t)  d(t) = recovery of original aignal Bandwidth : from 2 / Τ in y(t) becomes 2 / Τg in z(t)  Tg << T  / Tg >> 2 / T spread spectrum ! Applications : Police communications, GPS

Correlation of digital signals

Digital signal = linear combination of orthogonal pulses

Digital signal = linear combination of orthogonal pulses Elementary orthogonal pulse (duration Τ, amplitude 1,center t = 0) :

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) :

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 :

contributes to the total signal the component : Every other digit k places after the initial (or brfore for k<0) has center t = d + kT, where T = digit length, has contribution :

contributes to the total signal the component : Every other digit k places after the initial (or brfore for k<0) has center t = d + kT, where T = digit length, has contribution : Total digital signal (digits do not overlap) :

Digital signal :

Digital signal : Choice between the values +A and -A “random”, independently and with equal probability (= ½) : PRN = Pseudo Random Noise GPS ! Ak = random variable, x(t) = stochastic process (random function)

Digital signal : Choice between the values +A and -A “random”, independently and with equal probability (= ½) : PRN = Pseudo Random Noise GPS ! Ak = random variable, x(t) = stochastic process (random function) A stochastic process x(t) taking discrete values zi (i = 1, 2, ...) is characterized by the joined probabilities (for every n) :

Digital signal : Choice between the values +A and -A “random”, independently and with equal probability (= ½) : PRN = Pseudo Random Noise GPS ! Ak = random variable, x(t) = stochastic process (random function) A stochastic process x(t) taking discrete values zi (i = 1, 2, ...) is characterized by the joined probabilities (for every n) : mean function :

Digital signal : Choice between the values +A and -A “random”, independently and with equal probability (= ½) : PRN = Pseudo Random Noise GPS ! Ak = random variable, x(t) = stochastic process (random function) A stochastic process x(t) taking discrete values zi (i = 1, 2, ...) is characterized by the joined probabilities (for every n) : mean function : correlation function :

Digital signal : Choice between the values +A and -A “random”, independently and with equal probability (= ½) : PRN = Pseudo Random Noise GPS ! Ak = random variable, x(t) = stochastic process (random function) A stochastic process x(t) taking discrete values zi (i = 1, 2, ...) is characterized by the joined probabilities (for every n) : mean function : correlation function : covariance function :

Stochastic characteristics of PRN noise
(Pseudo Random Noise)

(Pseudo Random Noise) Digital signal (PRN code) as a stochastic process (random function)

(Pseudo Random Noise) Digital signal (PRN code) as a stochastic process (random function) Ak = random variables with possible values + A και -A, with equal probability and independent

(Pseudo Random Noise) Digital signal (PRN code) as a stochastic process (random function) Ak = random variables with possible values + A και -A, with equal probability and independent Probabilities : Probability ( Ak = +A ) = 1/2 Probability ( Αk = -A ) = 1/2

(Pseudo Random Noise) Digital signal (PRN code) as a stochastic process (random function) Ak = random variables with possible values + A και -A, with equal probability and independent Probabilities : Probability ( Ak = +A ) = 1/2 Probability ( Αk = -A ) = 1/2 Joint probabilities : Probability ( Ak = +A AND Aj = +A ) = ½ ½ = 1/4 Probability ( Αk = +A AND Aj = -A ) = ½ ½ = 1/4 Probability ( Αk = -A AND Aj = +A ) = ½ ½ = 1/4 Probability ( Αk = -A AND Aj = -A ) = ½ ½ = 1/4

(Pseudo Random Noise) Digital signal (PRN code) as a stochastic process (random function) Ak = random variables with possible values + A και -A, with equal probability and independent Probabilities : Probability ( Ak = +A ) = 1/2 Probability ( Αk = -A ) = 1/2 Joint probabilities : Probability ( Ak = +A AND Aj = +A ) = ½ ½ = 1/4 Probability ( Αk = +A AND Aj = -A ) = ½ ½ = 1/4 Probability ( Αk = -A AND Aj = +A ) = ½ ½ = 1/4 Probability ( Αk = -A AND Aj = -A ) = ½ ½ = 1/4 mean value: mAk  E{Ak} = 0

(Pseudo Random Noise) Digital signal (PRN code) as a stochastic process (random function) Ak = random variables with possible values + A και -A, with equal probability and independent Probabilities : Probability ( Ak = +A ) = 1/2 Probability ( Αk = -A ) = 1/2 Joint probabilities : Probability ( Ak = +A AND Aj = +A ) = ½ ½ = 1/4 Probability ( Αk = +A AND Aj = -A ) = ½ ½ = 1/4 Probability ( Αk = -A AND Aj = +A ) = ½ ½ = 1/4 Probability ( Αk = -A AND Aj = -A ) = ½ ½ = 1/4 mean value: mAk  E{Ak} = 0 variance: σAk2  E{(Ak-mAk)2} = E{Ak2} = A2

(Pseudo Random Noise) Digital signal (PRN code) as a stochastic process (random function) Ak = random variables with possible values + A και -A, with equal probability and independent Probabilities : Probability ( Ak = +A ) = 1/2 Probability ( Αk = -A ) = 1/2 Joint probabilities : Probability ( Ak = +A AND Aj = +A ) = ½ ½ = 1/4 Probability ( Αk = +A AND Aj = -A ) = ½ ½ = 1/4 Probability ( Αk = -A AND Aj = +A ) = ½ ½ = 1/4 Probability ( Αk = -A AND Aj = -A ) = ½ ½ = 1/4 mean value: mAk  E{Ak} = 0 variance: σAk2  E{(Ak-mAk)2} = E{Ak2} = A2 covariance: σAkAj  E{(Ak-mAk)(Aj-mAj)} = E{AkAj} = 0 (k  j)

Determination of the stochastic characteristics of PRN code

Digital signal (PRN code) as a stochastic process (random function)

Digital signal (PRN code) as a stochastic process (random function) Ak = randomvariables with possible values x = ± A, with equal probability and independent:

Digital signal (PRN code) as a stochastic process (random function) Ak = randomvariables with possible values x = ± A, with equal probability and independent: Probabilities :

Digital signal (PRN code) as a stochastic process (random function) Ak = randomvariables with possible values x = ± A, with equal probability and independent: Probabilities : Joint probabilities ( x = ± A, y = ± A) :

Digital signal (PRN code) as a stochastic process (random function) Ak = randomvariables with possible values x = ± A, with equal probability and independent: Probabilities : Joint probabilities ( x = ± A, y = ± A) : mean value:

Digital signal (PRN code) as a stochastic process (random function) Ak = randomvariables with possible values x = ± A, with equal probability and independent: Probabilities : Joint probabilities ( x = ± A, y = ± A) : mean value: variance:

Digital signal (PRN code) as a stochastic process (random function) Ak = randomvariables with possible values x = ± A, with equal probability and independent: Probabilities : Joint probabilities ( x = ± A, y = ± A) : mean value: variance: covariance:

where :

where : The stochastic process x(t) has mean value

where : The stochastic process x(t) has mean value because

where : The stochastic process x(t) has mean value because and covariance function

where : The stochastic process x(t) has mean value because and covariance function Therefore : when t1 and t2 belong to the same digit k

where : The stochastic process x(t) has mean value because and covariance function Therefore : when t1 and t2 belong to the same digit k when t1 and t2 belong to different digits k1  k2

where : The stochastic process x(t) has mean value because and covariance function Therefore : when t1 and t2 belong to the same digit k when t1 and t2 belong to different digits k1  k2 To determine whether t1 and t2 are within the same digit since R(t1, t2) = R(t2, t1), we shall examine only the case t1 < t2 (otherwise we swap t1 and t2)

There exist 3 cases

There exist 3 cases (Α) τ = t2 – t1 > T t1, t2 always belong to different digits T-τ τ δ t1 t2

There exist 3 cases (Α) τ = t2 – t1 > T t1, t2 always belong to different digits (Β) τ = t2 – t1 < T t1, t2 belong to different digits T-τ τ δ t1 t2 T-τ τ δ t1 t2

There exist 3 cases (Α) τ = t2 – t1 > T t1, t2 always belong to different digits (Β) τ = t2 – t1 < T t1, t2 belong to different digits (C) τ = t2 – t1 < T t1, t2 belong to the same digit T-τ τ δ t1 t2 T-τ τ δ t1 t2 T-τ τ δ t1 t2

There exist 3 cases (Α) τ = t2 – t1 > T t1, t2 always belong to different digits (Β) τ = t2 – t1 < T t1, t2 belong to different digits (C) τ = t2 – t1 < T t1, t2 belong to the same digit δ + τ > Τ  T-τ τ δ t1 t2 T-τ τ δ t1 t2 T-τ τ δ t1 t2

There exist 3 cases (Α) τ = t2 – t1 > T t1, t2 always belong to different digits (Β) τ = t2 – t1 < T t1, t2 belong to different digits (C) τ = t2 – t1 < T t1, t2 belong to the same digit δ + τ > Τ  δ > T-τ T-τ τ δ t1 t2 T-τ τ δ t1 t2 T-τ τ δ t1 t2

There exist 3 cases (Α) τ = t2 – t1 > T t1, t2 always belong to different digits (Β) τ = t2 – t1 < T t1, t2 belong to different digits (C) τ = t2 – t1 < T t1, t2 belong to the same digit δ + τ > Τ  δ > T-τ δ + τ < Τ  T-τ τ δ t1 t2 T-τ τ δ t1 t2 T-τ τ δ t1 t2

There exist 3 cases (Α) τ = t2 – t1 > T t1, t2 always belong to different digits (Β) τ = t2 – t1 < T t1, t2 belong to different digits (C) τ = t2 – t1 < T t1, t2 belong to the same digit δ + τ > Τ  δ > T-τ δ + τ < Τ  δ < T-τ T-τ τ δ t1 t2 T-τ τ δ t1 t2 T-τ τ δ t1 t2

There exist 3 cases (Α) τ = t2 – t1 > T t1, t2 always belong to different digits (Β) τ = t2 – t1 < T t1, t2 belong to different digits (C) τ = t2 – t1 < T t1, t2 belong to the same digit δ + τ > Τ  δ > T-τ δ + τ < Τ  δ < T-τ T-τ τ δ t1 t2 T-τ τ δ t1 t2 T-τ τ δ t1 t2 t1, t2 in different digits, the random variables and are independent and

There exist 3 cases (Α) τ = t2 – t1 > T t1, t2 always belong to different digits (Β) τ = t2 – t1 < T t1, t2 belong to different digits (C) τ = t2 – t1 < T t1, t2 belong to the same digit δ + τ > Τ  δ > T-τ δ + τ < Τ  δ < T-τ T-τ τ δ t1 t2 T-τ τ δ t1 t2 T-τ τ δ t1 t2 t1, t2 in different digits, the random variables and are independent and t1, t2 in neighboring digits, the random variables and are independent and

There exist 3 cases (Α) τ = t2 – t1 > T t1, t2 always belong to different digits (Β) τ = t2 – t1 < T t1, t2 belong to different digits (C) τ = t2 – t1 < T t1, t2 belong to the same digit δ + τ > Τ  δ > T-τ δ + τ < Τ  δ < T-τ T-τ τ δ t1 t2 T-τ τ δ t1 t2 T-τ τ δ t1 t2 t1, t2 in different digits, the random variables and are independent and t1, t2 in neighboring digits, the random variables and are independent and t1, t2 in the same digit, the random variables and are identical

B: T-τ < δ < Τ C: 0 < δ < T-τ

τ δ t1 t2 B: T-τ < δ < Τ C: 0 < δ < T-τ

τ δ t1 t2 B: T-τ < δ < Τ T-τ τ δ t1 t2 C: 0 < δ < T-τ

τ δ t1 t2 B: T-τ < δ < Τ T-τ τ δ t1 t2 C: 0 < δ < T-τ δ = random variable with homogeneous distribution in the interval (0 < δ < T ) :

τ δ t1 t2 B: T-τ < δ < Τ T-τ τ δ t1 t2 C: 0 < δ < T-τ δ = random variable with homogeneous distribution in the interval (0 < δ < T ) : Event C = (t1, t2 in the same digit)

τ δ t1 t2 B: T-τ < δ < Τ T-τ τ δ t1 t2 C: 0 < δ < T-τ δ = random variable with homogeneous distribution in the interval (0 < δ < T ) : Event B = (t1, t2 in different digits) Event C = (t1, t2 in the same digit)

Correlation of PRN code

Α2 Τ -Τ τ

Α2 Τ -Τ τ Corresponding spectral density : S(ω) Α2Τ ω

Correlation of a GPS signal with a receiver-produced copy

Correlation of a GPS signal with a receiver-produced copy
Assumption : The stochastic process x(t) is ergodic :

stochastic correlation correlation of power signal
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

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)

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) At the receiver : [ Multiplication with signal x(t – τ) with variable delay τ ] [integration]

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) At the receiver : [ Multiplication with signal x(t – τ) with variable delay τ ] [integration] (L = code length in digits)

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) At the receiver : [ Multiplication with signal x(t – τ) with variable delay τ ] [integration] (L = code length in digits) Variation of τ , until και

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) 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* !

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) 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

Correlation function

Correlation function + + –

Correlation function R(τ)+ + + –

Correlation function R(τ)+ + + – R(τ)-

Correlation function R(τ)+ + + – R(τ)- R(τ) = R(τ)+ + R(τ)-

= – R(τ)+ R(τ)- + + – R(τ) R(τ)+ | R(τ)- | R(τ) = R(τ)+ + R(τ)-
Correlation function R(τ)+ + + – R(τ)- = – R(τ) R(τ)+ | R(τ)- | R(τ) = R(τ)+ + R(τ)-

τ = - 5Τ /4 x(t) x(t-τ) x(t) x(t-τ) R(τ)+ R(τ)=R(τ)+ - R(τ)- R(τ) τ R(τ)- -T T

τ = - Τ x(t) x(t-τ) x(t) x(t-τ) R(τ)+ R(τ)=R(τ)+ - R(τ)- R(τ) τ R(τ)- -T T

τ = - Τ /4 x(t) x(t-τ) x(t) x(t-τ) R(τ)+ R(τ)=R(τ)+ - R(τ)- R(τ) τ R(τ)- -T T

τ = 0 x(t) x(t-τ) x(t) x(t-τ) R(τ)+ R(τ)=R(τ)+ - R(τ)- R(τ) τ R(τ)- -T T

τ = Τ /4 x(t) x(t-τ) x(t) x(t-τ) R(τ)+ R(τ)=R(τ)+ - R(τ)- R(τ) τ R(τ)- -T T

τ = 2Τ /4 x(t) x(t-τ) x(t) x(t-τ) R(τ)+ R(τ)=R(τ)+ - R(τ)- R(τ) τ R(τ)- -T T

τ = Τ x(t) x(t-τ) x(t) x(t-τ) R(τ)+ R(τ)=R(τ)+ - R(τ)- R(τ) τ R(τ)- -T T

Signals and Spectral Methods

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Signals and Spectral Methods

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