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The Basic Theory of Cyclostationary

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The Basic Theory of Cyclostationary

In the fields of statistical signal processing, statistics and time-series analysis, a cyclostationary process is a type of random process. They have applications to topics such as analyzing heart rate variability from electrocardiogram signals and digital communication systems that rely on pulse timing.

Cyclostationarity considers a stochastic process having a fixed point or points that do not move around as it evolves over time; this contrasts with other fixed-point processes such as polynomial chaos expansions where the fixed points are allowed to wander arbitrarily.

Statistical properties of cyclostationary processes are similar to those for periodically modulated signals. The power spectrum of the cyclo-stationary process, which is defined as the Fourier transform of its autocovariance function, has several peaks corresponding to frequencies with nonzero power.

However there are no nulls in the spectral estimates at zero frequency due to fixed points present in the signal. One way to resolve this “null problem” is by using wide sense stationarity instead of stationarity. If formula_1, where formula_2 denotes complex conjugate (and formula_3), then definitionally all moments exist and are finite; moreover that class of processes can be shown to be c-cyclostationary.

Approaches

There are several different approaches for the treatment of cyclo-stationary processes, corresponding to the behavior of the power spectrum at frequencies close to zero frequency. This includes:

(1) “wide sense stationarity” (WSS),

(2) weak/strong long range dependence,

(3) fractional Fourier transform or Fractional Cyclic Model (FCM).

The latter two methods involve evaluating spectral moments constructed from an appropriately defined wavelet. The FCM approach also involves computing a special family of ‘quasi periodograms’ that exhibit good behavior in the null problem region. Since these three schemes are closely related, it is often difficult to identify which method is most appropriate to use.

The spectral analysis of cyclostationary processes with zero frequency is not unique. For example, if formula_5 for all formula_6, then the power spectrum will have a single peak at zero frequency. However, the power spectrum in that case may exhibit large noise amplification near zero frequency.

The most common choice among these schemes is the WSS approach combined with quasi periodogram (WSS/QP) and Fractional Fourier transform (FCM). Although it is difficult to pick one method over another there are some theoretical results which help determine when each of these methods works well.

For this discussion we start by considering white noise processes formula_7 such that E [ j2πN(t) ] = 1 where formula_8 is the spectral density of the noise and N(t) corresponds to a white noise process.

Wide-sense Stationarity (WSS)

If we take F [ j2πN(t)] we see that it is real and nonzero for any real value of t. This implies that for any fixed value of u, F [ j2πN(u) ] will be nonzero and so will have a peak at u=0. This can be evaluated by taking the Fourier transform of both sides:

F [ j2πN(u) ] = E [ ej2πivu ] . Performing this integration gives us:

E [ ej2πN(u) ] + E [ ej2π-N(u) ] = 2E [ N(u) ]. So, taking the expectation value of both terms on the right hand side yields zero. This implies that the power spectrum will have a peak at u=0 which is WSS.

Weak/Strong Long-range Dependence (WLD/SRLD)

Let us start with some definitions:

A signal x(t), 0 < t ≤ T, is said to be weakly (resp., strongly) long-range dependent (wld (resp., srd)) if for any fixed positive integer β and nonnegative integers m and with 1 ≤ m ≤ T,

E [ | x(t) | βm ] is nonzero.

For any fixed integer k with 1 ≤ k ≤ K, consider the signal defined by formula_11 where for n = 0,1,…,K – 1 and N is very large. This process is called the fractional cyclic (FC) expansion of order K.

The power spectrum of FCM can be written as:

where FFT denotes fast Fourier transform and Rk(u) is a modified Bessel function of order k. That it should have a peak at u = 0 follows from the Weierstrass P-function theorem that implies that Rk(0) has infinitely many real zeroes.

This is a general result that can be proved for any integer k ≥ 1. The peak at u = 0 should be very sharp because Rk(0) has a large number of zeroes and hence the Fourier transform will have a narrow peak near zero frequency.

Conclusion

In conclusion, there are a few different ways to analyze cyclostationary processes. The most common approaches involve the WSS approach combined with quasi periodogram or Fractional Fourier transform. You can choose which one of these schemes is best for your needs based on certain theoretical results that determine when each scheme works well.

For instance, if you have a white noise process then power spectrum will be peaked at zero frequency and wide-sense stationarity applies. If weak long range dependence is present in your signal then this method may work better than strong long range dependency because it has sharper peaks near zero frequency due to more zeroes from Weierstrass P-function theorem.

Read More : Explain Cyclostationary Random Process

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