This chapter is from the book
This chapter is from the book
This chapter is from the book
Problems
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Construct a random process from any number of random variables that have a stationary autocorrelation and a nonstationary mean.
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Prove that the magnitude of a unit autocovariance of any random process is always less than or equal to 1.
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Fill in the blanks of the following table:
AUTOCORRELATION
MEAN, μh
AUTOCOVARIANCE
UNIT AUTOCOVARIANCE
S0 exp(–α|Δt|)
0
A
C
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Explain why the PSD for any random channel must be a function containing only real, nonnegative values.
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Below are eight real-valued functions. Each may be a valid autocorrelation function (A), a valid PSD (P), both (B), or neither (N). Mark each function with either A, P, B, or N, based on the properties of PSDs and autocorrelations. (Assume that the vertical line in the center is the zero point.)
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A WSS random process is said to be white if its PSD is a constant. Answer the following questions based on a time-varying random channel,
, that is white:
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Prove that the random Doppler process
is also WSS with respect to ω.
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For the process described in (a), write an expression for the time-domain PSD of
, 
.
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Use the Fourier transform to complete the following table of autocorrelation and PSD functions of one dependency:
AUTOCORRELATION
PSD
J0(2πΔt)cos(2πΔt)
6τ exp(–πτ)
cos(πω)u(1 – |ω|)
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If we desire to construct an autocorrelation function for a time-varying passband channel, h(t), we can use the following expression in terms of complex baseband channels,
[Bel63]:
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Derive the above expression from first principles. For the general case, what information besides
is needed to calculate ?
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If we know that the passband channel is WSS, what simplifications can we make to this expression?
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Consider a spectral autocorrelation for a time- and frequency-varying channel that takes the following form:
Use this form of the spectral autocorrelation to answer the following questions:
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If we desire only a narrowband channel analysis (no frequency selectivity), what is the criterion for a WSS time-varying channel?
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If we desire only a static channel analysis (no temporal selectivity), what is the criterion for a WSS frequency-varying channel?
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To simultaneously model frequency and temporal selectivity, what are the criteria for a WSS-US channel?
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Challenge: Derive criteria that produce a WSS time–frequency autocorrelation function (depends only on t1 – t2 and f1 – f2 with correlated scattering. This involves specifying a relationship between the functions T and W and their dependencies.
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Use the Fourier transform to complete the following table of autocorrelation and PSD functions of multiple dependencies:
AUTOCORRELATION
PSD
cos(πω) u(1 – |ω|) u(1 – |τ|) u(1 – |k|)
exp(–aω – bτ – ck) u(ω) u(τ) u(k)
exp(– j[aΔt + bΔf + cΔr])
exp (–Δf2 – 2|Δr|) (1 – |Δt|) u(1 – |Δt|)
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Calculate the RMS spreads and fading rate variances for the random channels described by the following:
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(needs a pair of answers)
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Not all random processes have well-defined RMS PSD widths. Consider a random channel with a triangular autocorrelation function,
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Answer the following questions based on this random process: -
Calculate the PSD for this process.
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Set up the integral for calculating an RMS Doppler spread. How does this integral evaluate?
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What does the result in (b) imply about the fading rate variance? What characteristics of a random process might imply such behavior?
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In the text, we stated that fading rate variance was a second-order statistic. Yet, upon inspection, this statistic is calculated from only a single sample of the differentiated random process. Explain why this is still considered a second-order statistic.
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Channel duality can be a powerful tool for understanding many signal processing techniques in wireless communications by constructing a duality analogy. For example, we can construct a sentence that uses duality to show similarity between the purposes of frequency-hopping and interleaving:
The technique of (frequency-hopping, interleaving) transmits sequential digital symbols in pseudorandom (frequency, time) slots so that bursts of errors are avoided in a (frequency, time) selective channel.
Construct a duality analogy for each of the following concepts. (This may require some research.)
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How multiple users share the common air interface in (TDMA, FDMA). T/FDMA is Time/Frequency Division Multiple Access.
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How fading is overcome using (frequency-hopping, selection antenna diversity).
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How increased spectral spreads lead to instability in (a linear equalizer, adaptive gain control).
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How to remove fading using (switched-beam antenna diversity, a rake equalizer). Hint: Switching beams in a linear antenna array is equivalent to selecting only a portion of a wavenumber spectrum.
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How digital symbols are modulated using (QAM, OFDM). QAM: quadrature amplitude modulation, OFDM: orthogonal frequency division multiplexing.
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Find two original concepts in digital modulation to create your own duality analogy.
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