# Finding System transfer function, $H(z)$ or, equivalently, its impulse response, $h[n]$

I just started on DSP and I have a question that I would like to ask.

I have a zero-mean uncorrelated wide sense stationary discrete-time random process {$x[n]$, $n$ is a set of integers}. $x[n]$ is input to a causal linear time-invariant system below, denoted by its transfer function $H(z)$.

$$x[n]{\longrightarrow}\boxed{H(z)}{\longrightarrow }y[n]$$

Since the output is a random process, suppose that its autocorrelation function, $R_y[k]$, is measured without approximation error.

From the information given, is there a method to identify the system transfer function, $H(z)$ or, equivalently, its impulse response, $h[n]$?

• i wanted to fix up some notation (first started with the answer which was necessary because something was falsely stated). we leave the "$k"$ in $R_x[k]$ because it's the lag, not the time index $n$. $$R_x[k] \triangleq \lim_{N \to \infty} \frac{1}{2N+1} \sum\limits_{n=-N}^{+N} x[n] \cdot x[n+k]$$ Nov 22 '16 at 17:37
• and if $x[n]$ is ergodic, then $$R_x[k] = \mathrm{E} \{ x[n] \cdot x[n+k] \}$$ where $\mathrm{E} \{ \cdot \}$ is the probabilistic Expectation value or Expected value and is derived from the probability density functions for the random process $x[n]$. if it's "ergodic" then all of the time averages equal probabilistic averages (or are taken to be equal). Nov 22 '16 at 17:43

From only the process statistics, you cannot derive the exact impulse response/transfer function. However, by the Wiener-Khinchin-Theorem you know, that the power spectrum density (PSD) of an input signal to an LTI system is transformed with the magnitude-square of the transfer function:

Let $S_x(\omega)$ be the PSD of the input and $S_y(\omega)$ be the PSD at the output of the system. Note that

\begin{align} S_x(\omega) &= \mathrm{DTFT} \{ R_x[n] \} \\ &= |X(e^{j\omega})|^2 \\ \end{align}

where $X(e^{j\omega})= \mathrm{DTFT} \{x[n]\}$ and similarly for $S_y(\omega)$.

Then, you have

$$S_y(\omega)=|H(e^{j\omega})|^2 S_x(\omega)$$

and you can at least calculate the magnitude of $H(e^{j\omega})$ by inverting the equation.