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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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”

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

10. Mobile Radio Propagation Channel
Vertical Dipole

11. 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

12. 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)

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

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

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

16. 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:

17. 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, ..

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

19. 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

20. 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 Jakes suggests to use 8 sinusoidal signals. ?
Approximation (orange) of the U-Doppler spectrum (Black)

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

22. Frequency Dispersion

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

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

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

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

27. Typical values of delay spread
Picocells GHz:
TRMS < 50 nsec nsec
Home 50 nsec
Shopping mall nsec
Railway station nsec
Office block nsec

28. 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

29. 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
Rayleigh
Rayleigh
Rayleigh
Rayleigh
Rayleigh
Rayleigh

30. COST 207 A sample fixed channel
Tap number Delay ( (ms) Amplitude r Level (dB) Phase ( (rad)
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

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

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

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

34. 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

35. 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 ).

36. 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 ).

37. 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

38. Mobile Radio Propagation Channel
Coherence Bandwidth

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

40. 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

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

42. Effects of fading on modulated radio signals

43. 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.

44. 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.

45. BER
BER for Ricean fading
calculate

46. Time Dispersion Revisited
The duration of fades

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

48. Mobile Radio Propagation Channel
Two state model

49. 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

50. 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)

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

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

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

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

55. Average nonfade duration
Mobile Radio Propagation Channel
Average nonfade duration
Calculate

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

57. Mobile Radio Propagation Channel
Optimal Packet length

58. Mobile Radio Propagation Channel
Optimal Packet length

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

60. Mobile Radio Propagation Channel
Average fade duration

61. 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

63. 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

64. 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/
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):

65. 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 ).

66. 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 ).

67. 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 ).

68. 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 ).

69. 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.