Cyclostationary stochastic process

A stochastic process ${\displaystyle x(t)}$ of mean ${\displaystyle \operatorname {E} [x(t)]}$ and autocorrelation function:

${\displaystyle R_{x}(t,\tau )=\operatorname {E} \{x(t+\tau )x^{*}(t)\},\,}$

where the star denotes complex conjugation, is said to be wide-sense cyclostationary with period ${\displaystyle T_{0}}$ if both ${\displaystyle \operatorname {E} [x(t)]}$ and ${\displaystyle R_{x}(t,\tau )}$ are cyclic in ${\displaystyle t}$ with period ${\displaystyle T_{0},}$ i.e.:^[2]

${\displaystyle \operatorname {E} [x(t)]=\operatorname {E} [x(t+T_{0})]{\text{ for all }}t}$

${\displaystyle R_{x}(t,\tau )=R_{x}(t+T_{0};\tau ){\text{ for all }}t,\tau .}$

The autocorrelation function is thus periodic in t and can be expanded in Fourier series:

${\displaystyle R_{x}(t,\tau )=\sum _{n=-\infty }^{\infty }R_{x}^{n/T_{0}}(\tau )e^{j2\pi {\frac {n}{T_{0}}}t}}$

where ${\displaystyle R_{x}^{n/T_{0}}(\tau )}$ is called cyclic autocorrelation function and equal to:

${\displaystyle R_{x}^{n/T_{0}}(\tau )={\frac {1}{T_{0}}}\int _{-T_{0}/2}^{T_{0}/2}R_{x}(t,\tau )e^{-j2\pi {\frac {n}{T_{0}}}t}\mathrm {d} t.}$

The frequencies ${\displaystyle n/T_{0},\,n\in \mathbb {Z} ,}$ are called cycle frequencies.

Wide-sense stationary processes are a special case of cyclostationary processes with only ${\displaystyle R_{x}^{0}(\tau )\neq 0}$.

Cyclostationary time series

A signal that is just a function of time and not a sample path of a stochastic process can exhibit cyclostationarity properties in the framework of the fraction-of-time point of view. This way, the cyclic autocorrelation function can be defined by:^[2]

${\displaystyle {\widehat {R}}_{x}^{n/T_{0}}(\tau )=\lim _{T\rightarrow +\infty }{\frac {1}{T}}\int _{-T/2}^{T/2}x(t+\tau )x^{*}(t)e^{-j2\pi {\frac {n}{T_{0}}}t}\mathrm {d} t.}$

If the time-series is a sample path of a stochastic process it is ${\displaystyle R_{x}^{n/T_{0}}(\tau )=\operatorname {E} \left[{\widehat {R}}_{x}^{n/T_{0}}(\tau )\right]}$. If the signal is further cycloergodic,^[3] all sample paths exhibit the same cyclic time-averages with probability equal to 1 and thus ${\displaystyle R_{x}^{n/T_{0}}(\tau )={\widehat {R}}_{x}^{n/T_{0}}(\tau )}$ with probability 1.

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:

${\displaystyle S_{x}^{\alpha }(f)=\int _{-\infty }^{+\infty }R_{x}^{\alpha }(\tau )e^{-j2\pi f\tau }\mathrm {d} \tau .}$

The cyclic spectrum at zero 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.^[4]

The reason ${\displaystyle S_{x}^{\alpha }(f)}$ is called the spectral correlation density function is that it equals the limit, as filter bandwidth approaches zero, of the expected value of the product of the output of a one-sided bandpass filter with center frequency ${\displaystyle f+\alpha /2}$ and the conjugate of the output of another one-sided bandpass filter with center frequency ${\displaystyle f-\alpha /2}$, with both filter outputs frequency shifted to a common center frequency, such as zero, as originally observed and proved in.^[5]

For time series, the reason the cyclic spectral density function is called the spectral correlation density function is that it equals the limit, as filter bandwidth approaches zero, of the average over all time of the product of the output of a one-sided bandpass filter with center frequency ${\displaystyle f+\alpha /2}$ and the conjugate of the output of another one-sided bandpass filter with center frequency ${\displaystyle f-\alpha /2}$, with both filter outputs frequency shifted to a common center frequency, such as zero, as originally observed and proved in.^[6]

Example: linearly modulated digital signal

An example of cyclostationary signal is the linearly modulated digital signal :

${\displaystyle x(t)=\sum _{k=-\infty }^{\infty }a_{k}p(t-kT_{0})}$

where ${\displaystyle a_{k}\in \mathbb {C} }$ are i.i.d. random variables. The waveform ${\displaystyle p(t)}$, with Fourier transform ${\displaystyle P(f)}$, is the supporting pulse of the modulation.

By assuming ${\displaystyle \operatorname {E} [a_{k}]=0}$ and ${\displaystyle \operatorname {E} [|a_{k}|^{2}]=\sigma _{a}^{2}}$, the auto-correlation function is:

${\displaystyle {\begin{aligned}R_{x}(t,\tau )&=\operatorname {E} [x(t+\tau )x^{*}(t)]\\[6pt]&=\sum _{k,n}\operatorname {E} [a_{k}a_{n}^{*}]p(t+\tau -kT_{0})p^{*}(t-nT_{0})\\[6pt]&=\sigma _{a}^{2}\sum _{k}p(t+\tau -kT_{0})p^{*}(t-kT_{0}).\end{aligned}}}$

The last summation is a periodic summation, hence a signal periodic in t. This way, ${\displaystyle x(t)}$ is a cyclostationary signal with period ${\displaystyle T_{0}}$ and cyclic autocorrelation function:

${\displaystyle {\begin{aligned}R_{x}^{n/T_{0}}(\tau )&={\frac {1}{T_{0}}}\int _{-T_{0}/2}^{T_{0}/2}R_{x}(t,\tau )e^{-j2\pi {\frac {n}{T_{0}}}t}\,\mathrm {d} t\\[6pt]&={\frac {1}{T_{0}}}\int _{-T_{0}/2}^{T_{0}/2}\sigma _{a}^{2}\sum _{k=-\infty }^{\infty }p(t+\tau -kT_{0})p^{*}(t-kT_{0})e^{-j2\pi {\frac {n}{T_{0}}}t}\mathrm {d} t\\[6pt]&={\frac {\sigma _{a}^{2}}{T_{0}}}\sum _{k=-\infty }^{\infty }\int _{-T_{0}/2-kT_{0}}^{T_{0}/2-kT_{0}}p(\lambda +\tau )p^{*}(\lambda )e^{-j2\pi {\frac {n}{T_{0}}}(\lambda +kT_{0})}\mathrm {d} \lambda \\[6pt]&={\frac {\sigma _{a}^{2}}{T_{0}}}\int _{-\infty }^{\infty }p(\lambda +\tau )p^{*}(\lambda )e^{-j2\pi {\frac {n}{T_{0}}}\lambda }\mathrm {d} \lambda \\[6pt]&={\frac {\sigma _{a}^{2}}{T_{0}}}p(\tau )*\left\{p^{*}(-\tau )e^{j2\pi {\frac {n}{T_{0}}}\tau }\right\}.\end{aligned}}}$

with ${\displaystyle *}$ indicating convolution. The cyclic spectrum is:

${\displaystyle S_{x}^{n/T_{0}}(f)={\frac {\sigma _{a}^{2}}{T_{0}}}P(f)P^{*}\left(f-{\frac {n}{T_{0}}}\right).}$

Typical raised-cosine pulses adopted in digital communications have thus only ${\displaystyle n=-1,0,1}$ non-zero cyclic frequencies.

This same result can be obtained for the non-stochastic time series model of linearly modulated digital signals in which expectation is replaced with infinite time average, but this requires a somewhat modified mathematical method as originally observed and proved in.^[7]

Angle-time cyclostationarity of mechanical signals

Mechanical signals produced by rotating or reciprocating machines are remarkably well modelled as cyclostationary processes. The cyclostationary family accepts all signals with hidden periodicities, either of the additive type (presence of tonal components) or multiplicative type (presence of periodic modulations). This happens to be the case for noise and vibration produced by gear mechanisms, bearings, internal combustion engines, turbofans, pumps, propellers, etc. The explicit modelling of mechanical signals as cyclostationary processes has been found useful in several applications, such as in noise, vibration, and harshness (NVH) and in condition monitoring.^[19] In the latter field, cyclostationarity has been found to generalize the envelope spectrum, a popular analysis technique used in the diagnostics of bearing faults.

One peculiarity of rotating machine signals is that the period of the process is strictly linked to the angle of rotation of a specific component – the “cycle” of the machine. At the same time, a temporal description must be preserved to reflect the nature of dynamical phenomena that are governed by differential equations of time. Therefore, the angle-time autocorrelation function is used,

${\displaystyle R_{x}(\theta ,\tau )=\operatorname {E} \{x(t(\theta )+\tau )x^{*}(t(\theta ))\},\,}$

where ${\displaystyle \theta }$ stands for angle, ${\displaystyle t(\theta )}$ for the time instant corresponding to angle ${\displaystyle \theta }$ and ${\displaystyle \tau }$ for time delay. Processes whose angle-time autocorrelation function exhibit a component periodic in angle, i.e. such that ${\displaystyle R_{x}(\theta$ ;\tau )} has a non-zero Fourier-Bohr coefficient for some angular period ${\displaystyle \Theta }$, are called (wide-sense) angle-time cyclostationary. The double Fourier transform of the angle-time autocorrelation function defines the order-frequency spectral correlation,

${\displaystyle S_{x}^{\alpha }(f)=\lim _{S\rightarrow +\infty }{\frac {1}{S}}\int _{-S/2}^{S/2}\int _{-\infty }^{+\infty }R_{x}(\theta ,\tau )e^{-j2\pi f\tau }e^{-j2\pi \alpha {\frac {\theta }{\Theta }}}\,\mathrm {d} \tau \,\mathrm {d} \theta }$

where ${\displaystyle \alpha }$ is an order (unit in events per revolution) and ${\displaystyle f}$ a frequency (unit in Hz).

For constant speed of rotation, ${\displaystyle \omega }$, angle is proportional to time, ${\displaystyle \theta =\omega t}$. Consequently, the angle-time autocorrelation is simply a cyclicity-scaled traditional autocorrelation; that is, the cycle frequencies are scaled by ${\displaystyle \omega }$. On the other hand, if the speed of rotation changes with time, then the signal is no longer cyclostationary (unless the speed varies periodically). Therefore, it is not a model for cyclostationary signals. It is not even a model for time-warped cyclostationarity, although it can be a useful approximation for sufficiently slow changes in speed of rotation. ^[20]