Properties of the Mobile Radio Propagation Channel

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Presentation transcript:

Properties of the Mobile Radio Propagation Channel Jean-Paul M.G. Linnartz Department Head CoSiNe Nat.Lab., Philips Research

Statistical Description of Multipath Fading Mobile Radio Propagation Channel Statistical Description of Multipath Fading The basic Rayleigh / Ricean model gives the PDF of envelope But: how fast does the signal fade? How wide in bandwidth are fades? Typical system engineering questions: What is an appropriate packet duration, to avoid fades? How much ISI will occur? For frequency diversity, how far should one separate carriers? How far should one separate antennas for diversity? What is good a interleaving depth? What bit rates work well? Why can't I connect an ordinary modem to a cellular phone? The models discussed in the following sheets will provide insight in these issues

The Mobile Radio Propagation Channel A wireless channel exhibits severe fluctuations for small displacements of the antenna or small carrier frequency offsets. Amplitude Frequency Time Channel Amplitude in dB versus location (= time*velocity) and frequency

Time Dispersion vs Frequency Dispersion Mobile Radio Propagation Channel Time Dispersion vs Frequency Dispersion Time Dispersion Frequency Dispersion Time Domain Channel variations Delay spread Interpretation Fast Fading InterSymbol Interference Correlation Distance Channel equalization Frequency Doppler spectrum Frequency selective fading Domain Intercarrier Interf. Coherence bandwidth Interpretation

Two distinct mechanisms Mobile Radio Propagation Channel Two distinct mechanisms 1.) Time dispersion Time variations of the channel are caused by motion of the antenna Channel changes every half a wavelength Moving antenna gives Doppler spread Fast fading requires short packet durations, thus high bit rates Time dispersion poses requirements on synchronization and rate of convergence of channel estimation Interleaving may help to avoid burst errors 2.) Frequency dispersion Delayed reflections cause intersymbol interference Channel Equalization may be needed. Frequency selective fading Multipath delay spreads require long symbol times Frequency diversity or spread spectrum may help

Time dispersion of narrowband signal (single frequency) Mobile Radio Propagation Channel Time dispersion of narrowband signal (single frequency) Transmit: cos(2p fc t) Receive: I(t) cos(2p fc t) + Q(t) sin(2p fc t) = R(t) cos(2p fc t + f) I-Q phase trajectory As a function of time, I(t) and Q(t) follow a random trajectory through the complex plane Intuitive conclusion: Deep amplitude fades coincide with large phase rotations Animate

Doppler shift and Doppler spread Mobile Radio Propagation Channel Doppler shift and Doppler spread All reflected waves arrive from a different angle All waves have a different Doppler shift The Doppler shift of a particular wave is Maximum Doppler shift: fD = fc v / c Joint Signal Model Infinite number of waves First find distribution of angle of arrival, then compute distribution of Doppler shifts Uniform distribution of angle of arrival f: fF(f) = 1/2p Line spectrum goes into continuous spectrum Calculate

Mobile Radio Propagation Channel Doppler Spectrum If one transmits a sinusoid, … what are the frequency components in the received signal? Power density spectrum versus received frequency Probability density of Doppler shift versus received frequency The Doppler spectrum has a characteristic U-shape. Note the similarity with sampling a randomly-phased sinusoid No components fall outside interval [fc- fD, fc+ fD] Components of + fD or -fD appear relatively often Fades are not entirely “memory-less”

Derivation of Doppler Spectrum Mobile Radio Propagation Channel Derivation of Doppler Spectrum

Mobile Radio Propagation Channel Vertical Dipole

How do systems handle Doppler Spreads? Mobile Radio Propagation Channel How do systems handle Doppler Spreads? Analog Carrier frequency is low enough to avoid problems (random FM) GSM Channel bit rate well above Doppler spread TDMA during each bit / burst transmission the channel is fairly constant. Receiver training/updating during each transmission burst Feedback frequency correction DECT Intended to pedestrian use: only small Doppler spreads are to be anticipated for. IS95 Downlink: Pilot signal for synchronization and channel estimation Uplink: Continuous tracking of each signal

Autocorrelation of the signal Mobile Radio Propagation Channel Autocorrelation of the signal We now know the Doppler spectrum, but ... how fast does the channel change? Wiener-Kinchine Theorem: Power density spectrum of a random signal is the Fourier Transform of its autocorrelation Inverse Fourier Transform of Doppler spectrum gives autocorrelation of I(t), or of Q(t)

Derivation of Autocorrelation of I-Q components Mobile Radio Propagation Channel Derivation of Autocorrelation of I-Q components

For uniform angle of arrival Mobile Radio Propagation Channel For uniform angle of arrival

Relation between I and Q phase Mobile Radio Propagation Channel Relation between I and Q phase

PDF of the real amplitude R Mobile Radio Propagation Channel PDF of the real amplitude R For the amplitude r1 and r2 R2 = I2 + Q2 Note: get more elegant expressions for complex amplitude H Correlation: NB:

Autocorrelation of amplitude R2 = I2 + Q2 Mobile Radio Propagation Channel Autocorrelation of amplitude R2 = I2 + Q2 The solution is known as the hypergeometric function F(a,b;c;z) or, in good approximation, ..

Autocorrelation of amplitude R2 = I2 + Q2 Mobile Radio Propagation Channel Autocorrelation of amplitude R2 = I2 + Q2

Amplitude r(t0) and Derivative r’(t0) are uncorrelated Mobile Radio Propagation Channel Amplitude r(t0) and Derivative r’(t0) are uncorrelated J0() Correlation is 0 for t = 0

Simulation of multipath channels Jakes' Simulator for narrowband channel generate the two bandpass “noise” components by adding many sinusoidal signals. Their frequencies are non- uniformly distributed to approximate the typical U-shaped Doppler spectrum. N Frequency components are taken at 2p i fi = fm cos -------- 2(2N+1) with i = 1, 2, .., N All amplitudes are taken equal to unity. One component at the maximum Doppler shift is also added, but at amplitude of 1/sqrt(2), i.e., at about 0.707 . Jakes suggests to use 8 sinusoidal signals. ? Approximation (orange) of the U-Doppler spectrum (Black)

How to handle fast multipath fading? Mobile Radio Propagation Channel How to handle fast multipath fading?

Frequency Dispersion

Frequency Dispersion Frequency dispersion is caused by the delay spread of the channel Frequency dispersion has no relation to the velocity of the antenna

Frequency Dispersion: Delay Profile Mobile Radio Propagation Channel Frequency Dispersion: Delay Profile

RMS Delay Spread and Maximum delay spread Mobile Radio Propagation Channel RMS Delay Spread and Maximum delay spread

Typical Values of Delay Spreads Mobile Radio Propagation Channel Typical Values of Delay Spreads

Typical values of delay spread Picocells 1 .. 2 GHz: TRMS < 50 nsec - 300 nsec Home 50 nsec Shopping mall 100 - 200 nsec Railway station 200 - 450 nsec Office block 100 - 400 nsec

Typical Delay Profiles Mobile Radio Propagation Channel Typical Delay Profiles For a bandlimited signal, one may sample the delay profile. This give a tapped delay line model for the channel, with constant delay times

COST 207 Typical Urban Reception (TU6) COST 207 describes typical channel characteristics for over transmit bandwidths of 10 to 20 MHz around 900MHz. TU-6 models the terrestrial propagation in an urban area. It uses 6 resolvable paths COST 207 profiles were adapted to mobile DVB-T reception in the E.U. Motivate project. Tap number Delay (us) Power (dB) Fading model 1 0.0 -3 Rayleigh 2 0.2 0 Rayleigh 3 0.5 -2 Rayleigh 4 1.6 -6 Rayleigh 5 2.3 -8 Rayleigh 6 5.0 -10 Rayleigh

COST 207 A sample fixed channel Tap number Delay ( (ms) Amplitude r Level (dB) Phase ( (rad) 1 0.050 0.36 -8.88 -2.875 2 0.479 1 0 0 3 0.621 0.787 -2.09 2.182 4 1.907 0.587 -4.63 -0.460 5 2.764 0.482 -6.34 -2.616 6 3.193 0.451 -6.92 2.863 COST 207 Digital land mobile radio Communications, final report, September 1988. European Project AC 318 Motivate: Deliverable 06: Reference Receiver Conditions for Mobile Reception, January 2000

How do systems handle delay spreads? Mobile Radio Propagation Channel How do systems handle delay spreads?

Correlation of Fading vs. Frequency Separation Mobile Radio Propagation Channel Correlation of Fading vs. Frequency Separation

Inphase and Quadrature-Phase Components Mobile Radio Propagation Channel Inphase and Quadrature-Phase Components

Mobile Radio Propagation Channel Multipath Channel Transmit signal s(t) Received signal r(t) = an gn (t) + n(t), Channel model: Iw reflected waves have the following properties: Di is the amplitude Ti is the delay TO DO make math consistent The Wide Sense Stationary Uncorrelated Scattering (WSSUS) multipath channel is modeled as a collection of Iw reflected waves. Each wave has its particular Doppler frequency offset i, path delay Ti and amplitude Di, each of which is assumed to be constant. That is, we make the common assumption that the time-varying nature of the channel arises from the accumulation of multiple components. Due to motion of the antenna at constant velocity, each component has a linearly increasing phase offset, though all with a different slope. The Doppler offset i = 2fi lies within the Doppler spread -2f  i  2f, with f = vfc/c the maximum Doppler shift. Here v is the velocity of the mobile antenna, c is the speed of light. The carrier frequency is 2fc = c. The received signal r(t) consists of the composition of all reflected waves. Detection of the signal at subcarrier m occurs by multiplication with the m-th subcarrier frequency, thus with exp{-jct-jmst +m} during an appropriately chosen interval Ts. We take phase compensation m = 0. Vector Y describes the outputs of the FFT at the receiver, with Y = [y0, y1, .., yN-1]T. We assuming rectangular pulses of duration Ts. We write ym = nanm,nTs where m,n can be interpreted as the ‘leakage’ for a signal transmitted at subcarrier n and received at subcarrier m. This result can be interpreted as sampling in frequency domain: the Iw multipath channel contributions appear weighed according to their individual Doppler offset i. It confirms that due to the Doppler shifts, the detected signal ym contains contributions from all n subcarrier signals, not only from m = n. All m,n‘s with m  n lead to ICI, with amplitudes weighed by sinc( + i /s ). Signal parameters: an is the data c is the carrier frequency

Doppler Multipath Channel Mobile Radio Propagation Channel Doppler Multipath Channel Correlation between p-th and q-th derivative TO DO make math consistent The Wide Sense Stationary Uncorrelated Scattering (WSSUS) multipath channel is modeled as a collection of Iw reflected waves. Each wave has its particular Doppler frequency offset i, path delay Ti and amplitude Di, each of which is assumed to be constant. That is, we make the common assumption that the time-varying nature of the channel arises from the accumulation of multiple components. Due to motion of the antenna at constant velocity, each component has a linearly increasing phase offset, though all with a different slope. The Doppler offset i = 2fi lies within the Doppler spread -2f  i  2f, with f = vfc/c the maximum Doppler shift. Here v is the velocity of the mobile antenna, c is the speed of light. The carrier frequency is 2fc = c. The received signal r(t) consists of the composition of all reflected waves. Detection of the signal at subcarrier m occurs by multiplication with the m-th subcarrier frequency, thus with exp{-jct-jmst +m} during an appropriately chosen interval Ts. We take phase compensation m = 0. Vector Y describes the outputs of the FFT at the receiver, with Y = [y0, y1, .., yN-1]T. We assuming rectangular pulses of duration Ts. We write ym = nanm,nTs where m,n can be interpreted as the ‘leakage’ for a signal transmitted at subcarrier n and received at subcarrier m. This result can be interpreted as sampling in frequency domain: the Iw multipath channel contributions appear weighed according to their individual Doppler offset i. It confirms that due to the Doppler shifts, the detected signal ym contains contributions from all n subcarrier signals, not only from m = n. All m,n‘s with m  n lead to ICI, with amplitudes weighed by sinc( + i /s ).

Doppler Multipath Channel Mobile Radio Propagation Channel Doppler Multipath Channel TO DO make math consistent The Wide Sense Stationary Uncorrelated Scattering (WSSUS) multipath channel is modeled as a collection of Iw reflected waves. Each wave has its particular Doppler frequency offset i, path delay Ti and amplitude Di, each of which is assumed to be constant. That is, we make the common assumption that the time-varying nature of the channel arises from the accumulation of multiple components. Due to motion of the antenna at constant velocity, each component has a linearly increasing phase offset, though all with a different slope. The Doppler offset i = 2fi lies within the Doppler spread -2f  i  2f, with f = vfc/c the maximum Doppler shift. Here v is the velocity of the mobile antenna, c is the speed of light. The carrier frequency is 2fc = c. The received signal r(t) consists of the composition of all reflected waves. Detection of the signal at subcarrier m occurs by multiplication with the m-th subcarrier frequency, thus with exp{-jct-jmst +m} during an appropriately chosen interval Ts. We take phase compensation m = 0. Vector Y describes the outputs of the FFT at the receiver, with Y = [y0, y1, .., yN-1]T. We assuming rectangular pulses of duration Ts. We write ym = nanm,nTs where m,n can be interpreted as the ‘leakage’ for a signal transmitted at subcarrier n and received at subcarrier m. This result can be interpreted as sampling in frequency domain: the Iw multipath channel contributions appear weighed according to their individual Doppler offset i. It confirms that due to the Doppler shifts, the detected signal ym contains contributions from all n subcarrier signals, not only from m = n. All m,n‘s with m  n lead to ICI, with amplitudes weighed by sinc( + i /s ).

Random Complex-Gaussian Amplitude Mobile Radio Propagation Channel Random Complex-Gaussian Amplitude Special case This defines the covariance matrix of subcarrier amplitudes at different frequencies This is used in OFDM for cahnnel estimation TO DO make math consistent

Mobile Radio Propagation Channel Coherence Bandwidth

Frequency and Time Dispersion Mobile Radio Propagation Channel Frequency and Time Dispersion

Scatter Function of a Multipath Mobile Channel Mobile Radio Propagation Channel Scatter Function of a Multipath Mobile Channel Gives power as function of · Doppler Shift (derived from angle of arrival f) · Excess Delay Example of a scatter plot Horizontal axes: · x-axis: Excess delay time · y-axis: Doppler shift Vertical axis · z-axis: received power

Mobile Radio Propagation Channel Freq. and time selective channels Special cases Zero displacement / motion:  = 0 Zero frequency separation: f = 0

Effects of fading on modulated radio signals

Effects of Multipath (I) Mobile Radio Propagation Channel Effects of Multipath (I) Frequency Time Narrowband Frequency Time Wideband Frequency Time OFDM Multipath reception and user mobility lead to channel behavior that is time and frequency dependent. In the following frequency-time diagrams, we depict where in frequency and when in time most of the bit energy is located. It is important to select a modulation scheme that is appropriate for the Doppler Spread and the Delay Spread of the channel. In narrowband communication (NB), the occupied bandwidth is small and the bit duration is long. The bit energy footprint is stretched in vertical direction. Intersymbol interference is neglectable is the bit duration is, say, ten times or more longer than the delay spread. However, due to multipath the entire signal (bandwidth) may vanish in a deep fade. Is fading is very fast, the channel may change during a bit transmission. This can occur if the Doppler spread is large compared to the transmit bandwidth. In wideband transmission (WB), i.e., transmission at high bit rate, the signal is unlikely to vanish completely in a deep fade, because its bandwidth is so wide that some components will be present at frequencies where the channel is good. Such frequency selective fading, however, leads to intersymbol interference, which must be handled, for instance by an adaptive equalizer. In MultiCarrier or OFDM systems, multiple bits are transmitted in parallel over multiple subcarriers. If one subcarrier is in a fade, the other may not. Error correction coding can be used to correct bit errors on faded subcarriers. Rapid fading (Doppler) may erode the orthogonality of closely spaced subcarriers.

Effects of Multipath (II) Mobile Radio Propagation Channel Effects of Multipath (II) Frequency Time + - DS-CDMA Frequency Time + - Hopping Time Frequency + - MC-CDMA Direct Sequence intentionally broadens the transmit spectrum by multiplying user bits with a fast random sequence. The wideband signal is unlikely to fade completely. If frequency selective fading occurs, the receiver sees a series of time-shifted versions of the chipped bit. A ‘rake’ receiver can separate the individual resolvable paths. Frequency Hopping: The carrier frequency is shifted frequently, to avoid fades or narrowband interference. While the signal may vanish during one hop (at a particular frequency), most likely other hops jump to frequencies where the channel is good. Error correction is applied to correct bits lost at faded frequencies. Slow and fast hopping methods differ in the relative duration spent at one frequency, compared to the user bit duration. Orthogonal MC-CDMA is a spectrum-spreading method which exploits advantages of OFDM and DS-CDMA. Each user bit is transmitted simultaneously (in parallel) over multiple subcarriers. This corresponds to a 90 degree rotation of a bit energy map in the frequency-time plane, when compared to DS-CDMA. While the signal may be lost some subcarriers, reliable communication is ensured through appropriate combining of energy from various subcarriers, taking into account the signal-to-interference ratios. In an interference-free environment, Maximum Ratio Diversity combining is the preferred receive strategy. In a noise free environment, equalization (i.e. making all subcarrier amplitudes equal) is preferred.

BER BER for Ricean fading calculate

Time Dispersion Revisited The duration of fades

Time Dispersion Revisited: Duration of Fades Mobile Radio Propagation Channel Time Dispersion Revisited: Duration of Fades

Mobile Radio Propagation Channel Two state model

Level crossings per second Mobile Radio Propagation Channel Level crossings per second Number of level crossing per sec is proportional to speed r' of crossing R (derivative r' = dr/dt) probability of r being in [R, R + dR]. This Prob = fR(r) dr

Derivation of Level Crossings per Second Mobile Radio Propagation Channel Derivation of Level Crossings per Second Random process r’ is derivative of the envelope r w.r.t. time Note: here we need the joint PDF; not the conditional PDF f(r’½r=R) We derive f(r’½r=R) from f(r’,r) and f(i1,i2,q1,q2)

Covariance matrix of (I, Q, I’, Q’) Mobile Radio Propagation Channel Covariance matrix of (I, Q, I’, Q’)

Mobile Radio Propagation Channel Joint PDF of R, R’, F, F’

Level Crossings per Second Mobile Radio Propagation Channel Level Crossings per Second Calculate

Average Fade / Nonfade Duration Mobile Radio Propagation Channel Average Fade / Nonfade Duration Calculate

Average nonfade duration Mobile Radio Propagation Channel Average nonfade duration Calculate

How to handle long fades when the user is stationary? Mobile Radio Propagation Channel How to handle long fades when the user is stationary?

Mobile Radio Propagation Channel Optimal Packet length

Mobile Radio Propagation Channel Optimal Packet length

Derivation of Optimal Packet length Mobile Radio Propagation Channel Derivation of Optimal Packet length Calculate

Mobile Radio Propagation Channel Average fade duration

Conclusion The multipath channel is characterized by two effects: Time and Frequency Dispersion Time Dispersion effects are proportional to speed and carrier frequency System designer need to anticipate for channel anomalies

Statistical properties of a Rayleigh signal Parameter Probability Distribution I(t1) and Q(t1) i.i.d. Gaussian, Zero mean I(t1) and Q(t2) Correlated jointly Gaussian I(t2) given I(t1) Gaussian Amplitude r Rayleigh r(t2) given r(t1) Ricean r’(t1) derivative r Gaussian, Independent of amplitude Phase Uniform Derivative of Phase Gaussian, Dependent on amplitude Power Exponential The nice thing about jointly Gaussian r.v.s is that the covariance matrix fully describes the behavior

Taylor Expansion of Amplitude Rewrite the Channel Model as follows Tayler expansion of the amplitude gn(t) = gn(0)+ gn(1) (t-t) + gn(2) (t-t)2/2 + .. . gn(q) : the q-th derivative of amplitude wrt time, at instant t = t. gn(p) is a complex Gaussian random variable To address frequency separation nws (later):

Doppler Multipath Channel Mobile Radio Propagation Channel Doppler Multipath Channel Simplified baseband model for narrowband linear modulation Received signal r(t) = an g(t) + n(t), with time-varying channel amplitude g(t) Channel model: Iw reflected waves, each has the following properties: Di is the amplitude I is the Doppler shift Ti is the path delay Note g(t) is not an impulse response; It is the channel amplification and phase Signal parameters: linear modulation an is the user data c is the carrier frequency n(t) is the noise The Wide Sense Stationary Uncorrelated Scattering (WSSUS) multipath channel is modeled as a collection of Iw reflected waves. Each wave has its particular Doppler frequency offset i, path delay Ti and amplitude Di, each of which is assumed to be constant. That is, we make the common assumption that the time-varying nature of the channel arises from the accumulation of multiple components. Due to motion of the antenna at constant velocity, each component has a linearly increasing phase offset, though all with a different slope. The Doppler offset i = 2fi lies within the Doppler spread -2f  i  2f, with f = vfc/c the maximum Doppler shift. Here v is the velocity of the mobile antenna, c is the speed of light. The carrier frequency is 2fc = c. The received signal r(t) consists of the composition of all reflected waves. Detection of the signal at subcarrier m occurs by multiplication with the m-th subcarrier frequency, thus with exp{-jct-jmst +m} during an appropriately chosen interval Ts. We take phase compensation m = 0. Vector Y describes the outputs of the FFT at the receiver, with Y = [y0, y1, .., yN-1]T. We assuming rectangular pulses of duration Ts. We write ym = nanm,nTs where m,n can be interpreted as the ‘leakage’ for a signal transmitted at subcarrier n and received at subcarrier m. This result can be interpreted as sampling in frequency domain: the Iw multipath channel contributions appear weighed according to their individual Doppler offset i. It confirms that due to the Doppler shifts, the detected signal ym contains contributions from all n subcarrier signals, not only from m = n. All m,n‘s with m  n lead to ICI, with amplitudes weighed by sinc( + i /s ).

Doppler Multipath Channel Mobile Radio Propagation Channel Doppler Multipath Channel p-th Derivative gp(t) of the channel w.r.t. to time t All derivatives are gaussian rv’s if Iw   and Di i.i.d. The covariance matrix fully describes the statistical properties The Wide Sense Stationary Uncorrelated Scattering (WSSUS) multipath channel is modeled as a collection of Iw reflected waves. Each wave has its particular Doppler frequency offset i, path delay Ti and amplitude Di, each of which is assumed to be constant. That is, we make the common assumption that the time-varying nature of the channel arises from the accumulation of multiple components. Due to motion of the antenna at constant velocity, each component has a linearly increasing phase offset, though all with a different slope. The Doppler offset i = 2fi lies within the Doppler spread -2f  i  2f, with f = vfc/c the maximum Doppler shift. Here v is the velocity of the mobile antenna, c is the speed of light. The carrier frequency is 2fc = c. The received signal r(t) consists of the composition of all reflected waves. Detection of the signal at subcarrier m occurs by multiplication with the m-th subcarrier frequency, thus with exp{-jct-jmst +m} during an appropriately chosen interval Ts. We take phase compensation m = 0. Vector Y describes the outputs of the FFT at the receiver, with Y = [y0, y1, .., yN-1]T. We assuming rectangular pulses of duration Ts. We write ym = nanm,nTs where m,n can be interpreted as the ‘leakage’ for a signal transmitted at subcarrier n and received at subcarrier m. This result can be interpreted as sampling in frequency domain: the Iw multipath channel contributions appear weighed according to their individual Doppler offset i. It confirms that due to the Doppler shifts, the detected signal ym contains contributions from all n subcarrier signals, not only from m = n. All m,n‘s with m  n lead to ICI, with amplitudes weighed by sinc( + i /s ).

Doppler Multipath Channel Mobile Radio Propagation Channel Doppler Multipath Channel Correlation between p-th and q-th derivative The Wide Sense Stationary Uncorrelated Scattering (WSSUS) multipath channel is modeled as a collection of Iw reflected waves. Each wave has its particular Doppler frequency offset i, path delay Ti and amplitude Di, each of which is assumed to be constant. That is, we make the common assumption that the time-varying nature of the channel arises from the accumulation of multiple components. Due to motion of the antenna at constant velocity, each component has a linearly increasing phase offset, though all with a different slope. The Doppler offset i = 2fi lies within the Doppler spread -2f  i  2f, with f = vfc/c the maximum Doppler shift. Here v is the velocity of the mobile antenna, c is the speed of light. The carrier frequency is 2fc = c. The received signal r(t) consists of the composition of all reflected waves. Detection of the signal at subcarrier m occurs by multiplication with the m-th subcarrier frequency, thus with exp{-jct-jmst +m} during an appropriately chosen interval Ts. We take phase compensation m = 0. Vector Y describes the outputs of the FFT at the receiver, with Y = [y0, y1, .., yN-1]T. We assuming rectangular pulses of duration Ts. We write ym = nanm,nTs where m,n can be interpreted as the ‘leakage’ for a signal transmitted at subcarrier n and received at subcarrier m. This result can be interpreted as sampling in frequency domain: the Iw multipath channel contributions appear weighed according to their individual Doppler offset i. It confirms that due to the Doppler shifts, the detected signal ym contains contributions from all n subcarrier signals, not only from m = n. All m,n‘s with m  n lead to ICI, with amplitudes weighed by sinc( + i /s ).

Doppler Multipath Channel Mobile Radio Propagation Channel Doppler Multipath Channel The Wide Sense Stationary Uncorrelated Scattering (WSSUS) multipath channel is modeled as a collection of Iw reflected waves. Each wave has its particular Doppler frequency offset i, path delay Ti and amplitude Di, each of which is assumed to be constant. That is, we make the common assumption that the time-varying nature of the channel arises from the accumulation of multiple components. Due to motion of the antenna at constant velocity, each component has a linearly increasing phase offset, though all with a different slope. The Doppler offset i = 2fi lies within the Doppler spread -2f  i  2f, with f = vfc/c the maximum Doppler shift. Here v is the velocity of the mobile antenna, c is the speed of light. The carrier frequency is 2fc = c. The received signal r(t) consists of the composition of all reflected waves. Detection of the signal at subcarrier m occurs by multiplication with the m-th subcarrier frequency, thus with exp{-jct-jmst +m} during an appropriately chosen interval Ts. We take phase compensation m = 0. Vector Y describes the outputs of the FFT at the receiver, with Y = [y0, y1, .., yN-1]T. We assuming rectangular pulses of duration Ts. We write ym = nanm,nTs where m,n can be interpreted as the ‘leakage’ for a signal transmitted at subcarrier n and received at subcarrier m. This result can be interpreted as sampling in frequency domain: the Iw multipath channel contributions appear weighed according to their individual Doppler offset i. It confirms that due to the Doppler shifts, the detected signal ym contains contributions from all n subcarrier signals, not only from m = n. All m,n‘s with m  n lead to ICI, with amplitudes weighed by sinc( + i /s ).

Random Complex-Gaussian Amplitude Mobile Radio Propagation Channel Random Complex-Gaussian Amplitude It can be shown that for p + q is even and 0 for p + q is odd. This defines the covariance matrix of subcarrier amplitudes and derivatives, OFDM: allows system modeling and simulation between the input of the transmit I-FFT and output of the receive FFT.