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 \operatorname{E}[x(t)] and autocorrelation function: ::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 T_0 if both \operatorname{E}[x(t)] and R_x(t,\tau) are cyclic in t with period T_0, i.e.: all sample paths exhibit the same cyclic time-averages with probability equal to 1 and thus 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: ::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. The reason 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 f + \alpha /2 and the conjugate of the output of another one-sided bandpass filter with center frequency f - \alpha /2, with both filter outputs frequency shifted to a common center frequency, such as zero, as originally observed and proved in. 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 f + \alpha /2 and the conjugate of the output of another one-sided bandpass filter with center frequency f - \alpha /2, with both filter outputs frequency shifted to a common center frequency, such as zero, as originally observed and proved in.
Example: linearly modulated digital signal An example of cyclostationary signal is the
linearly modulated digital signal : ::x(t) = \sum_{k=-\infty}^{\infty} a_k p(t -kT_0) where a_k\in\mathbb{C} are
i.i.d. random variables. The waveform p(t), with Fourier transform P(f), is the supporting pulse of the modulation. By assuming \operatorname{E}[a_k] = 0 and \operatorname{E}[|a_k|^2]=\sigma_a^2, the auto-correlation function is: ::\begin{align} 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{align} The last summation is a
periodic summation, hence a signal periodic in
t. This way, x(t) is a cyclostationary signal with period T_0 and cyclic autocorrelation function: :: \begin{align} 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{align} with * indicating
convolution. The cyclic spectrum is: ::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 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. ==Cyclostationary models==