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Cyclo-stationary processes

A brief discussion of cyclostationary processes.
Prof. H.Amindavar complementary notes for the first session of "Advanced communications theory" course, Spring 2021

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Cyclo-stationary processes

  1. 1. Advanced Communication’s Theory Session 3 H.Amindavar February 2021 Cyclostationary Processes Definition A cyclostationary process is a signal having statistical properties that vary cyclically with time. There are two differing approaches to the treatment of cyclostationary processes. The probabilistic approach is to view measurements as an instance of a stochastic process. As an alternative, the The deterministic approach is to view the measurements as a single time series, from which a probability distribution for some event associated with the time series can be defined as the fraction of time that event occurs over the lifetime of the time series. In both approaches, the process or time series is said to be cyclostationary if and only if its associated probability distributions vary periodically with time. Wide-sense cyclostationary An important special case of cyclostationary signals is one that exhibits cyclostationarity in second-order statistics (e.g., the autocorrelation function). These are called wide-sense cyclostationary signals, and are analogous to wide-sense stationary processes. The exact definition differs depending on whether the signal is treated as a stochastic process or as a deterministic time series. Cyclostationary stochastic process A stochastic process x(t) of mean E[x(t)] and autocorrelation function: Rx(t, τ) = E{x(t + τ)x∗ (t)}, where the star denotes complex conjugation, is said to be wide-sense cyclostationary with period T0 if both E[x(t)] and Rx(t, τ) are cyclic in t , T0, E[x(t)] = E[x(t + T0)] for all t Rx(t, τ) = Rx(t + T0; τ) for all t, τ. The autocorrelation function is thus periodic in t and can be expanded in Fourier series: Rx(t, τ) = ∞ X n=−∞ Rn/T0 x (τ)ej2π n T0 t 1
  2. 2. Rn/T0 x (τ) is called cyclic autocorrelation function and equal to: Rn/T0 x (τ) = 1 T0 Z T0/2 −T0/2 Rx(t, τ)e−j2π n T0 t dt. n/T0, n ∈ Z, are called cyclic frequencies. Wide-sense stationary processes are a special case of cyclostationary processes with only R0 x(τ) 6= 0. Cyclostationary time series A signal that is just a function of time and not a sample path of a stochastic process can exhibit cyclostation- ary properties in the framework of the fraction-of-time point of view. This way, the cyclic autocorrelation function can be defined by: b Rn/T0 x (τ) = lim T →+∞ 1 T Z T/2 −T/2 x(t + τ)x∗ (t)e−j2π n T0 t dt. If the time-series is a sample path of a stochastic process it is Rn/T0 x (τ) = E h b Rn/T0 x (τ) i . If the signal is further ergodic, all sample paths exhibits the same time-average and thus Rn/T0 x (τ) = b Rn/T0 x (τ) in mean square error sense. Frequency domain behavior The Fourier transform of the cyclic autocorrelation function at cyclic frequency α is called cyclic spectrum or spectral correlation density function and is equal to: Sα x (f) = Z +∞ −∞ Rα x (τ)e−j2πfτ dτ. The cyclic spectrum at zeroth cyclic frequency is also called average power spectral density. For a Gaussian cyclostationary process, its rate distortion function can be expressed in terms of its cyclic spectrum. It is worth noting that a cyclostationary stochastic process x(t) with Fourier transform X(f) may have correlated frequency components spaced apart by multiples of 1/T0, since: E [X(f1)X∗ (f2)] = ∞ X n=−∞ Sn/T0 x (f1)δ f1 − f2 + n T0 Different frequencies f1,2 are always uncorrelated for a wide-sense stationary process since Sn/T0 x (f) 6= 0 only for n = 0. An example : linearly modulated digital signal An example of cyclostationary signal is the linearly modulated digital signal : x(t) = ∞ X k=−∞ akp(t − kT0) where ak ∈ C are i.i.d. random variables. The waveform p(t), with Fourier transform P (f), is the supporting pulse of the modulation. 2
  3. 3. W.A. Gardner / Cyclostationary time-series 33 (a) - I/2 0 + I/2 f +1 -I/2 0 + I/2 f +1 . I/2 0 + I/2 f •I/2 +1 0 + I/2 f Fig. 5. Cyclic spectrum magnitudes for PSK signals. (a) BPSK. (b) QPSK. (c) SQPSK. (d) MSK. Vol. 11, NO. 1, July 1986 Figure 1: Cyclic spectrum magnitude for PSK signals, α = n/T0 : (a)BPSK. (b)QPSK. (c)SQPSK. (d)MSK.1 By assuming E[ak] = 0 and E[|ak|2 ] = σ2 a , the autocorrelation function is: Rx(t, τ) = E[x(t + τ)x∗ (t)] = X k,n E[aka∗ n]p(t + τ − kT0)p∗ (t − nT0) = σ2 a X k p(t + τ − kT0)p∗ (t − kT0). The last summation is a periodic summation, hence a signal periodic in t. This way, x(t) is a cyclosta- tionary signal with period T0 and cyclic autocorrelation function: Rn/T0 x (τ) = 1 T0 Z T0 −T0 Rx(t, τ)e−j2π n T0 t dt = 1 T0 Z T0 −T0 σ2 a ∞ X k=−∞ p(t + τ − kT0)p∗ (t − kT0)e−j2π n T0 t dt = σ2 a T0 ∞ X k=−∞ Z T0−kT0 −T0−kT0 p(λ + τ)p∗ (λ)e−j2π n T0 (λ+kT0) dλ = σ2 a T0 Z ∞ −∞ p(λ + τ)p∗ (λ)e−j2π n T0 λ dλ = σ2 a T0 p(τ) ∗ n p∗ (−τ)ej2π n T0 τ o . 1Gardner, William A.(1986).”The spectral correlation theory of cyclostationary time-series”.Signal Processing.Elsevier.11: 13–36 3
  4. 4. The cyclic spectrum is: Sn/T0 x (f) = σ2 a T0 P (f)P ∗ f − n T0 . Typical raised-cosine pulses adopted in digital communications have thus only n = −1, 0, 1 non-zero cyclic frequencies. Figure 2: Impulse response of raised-cosine filter with various roll-off factors Figure 3: Frequency response of raised-cosine filter with various roll-off factors 4

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A brief discussion of cyclostationary processes. Prof. H.Amindavar complementary notes for the first session of "Advanced communications theory" course, Spring 2021

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