ECE538 Week 2: Autocorrelation and Cross-Correlation

This is where new content for the course starts. The purpose of this week was to understand autocorrelation, its properties, and its applicatiions.

The idea of autocorrelation is to see if a particular signal is embedded in a received signal. This is often used in radar applications, where the received signal is a time-delayed version of the original transmitted signal, with some noise added. Take the below example:

$$y[n]=\sum_{k=-\infty}^{\infty}a_kx[n-Dk]+w[n]$$

Notice the following:

$$z[n]=y[n]*x^*[-n]=\sum_{k=-\infty}^{\infty}a_kx[n-Dk]*x^*[-n]+w[n]*x^*[-n]$$$$z[n]=\sum_{k=-\infty}^{\infty}a_kx[n]*\delta(n-Dk)*x^*[-n]+w[n]*x^*[-n]$$$$z[n]=\sum_{k=-\infty}^{\infty}a_kx[n]*x^*[-n]*\delta(n-Dk)+w[n]*x^*[-n]$$$$z[n]=\sum_{k=-\infty}^{\infty}a_kr_{xx}[n]*\delta(n-Dk)+w[n]*x^*[-n]$$

Now, we define cross-correlation:

$$r_{xy}[l]=x[l]*y^*[-l]=\sum_{k=-\infty}^{\infty}x[k]y^*[k-l]$$

Autocorrelation occurs when $y=x$:

$$r_{xx}[l]=x[l]*x^*[-l]=\sum_{k=-\infty}^{\infty}x[k]x^*[k-l]$$

Thus,

$$z[n]=\sum_{k=-\infty}^{\infty}a_kr_{xx}[n-Dk]+r_{wx}$$

With the appropriate choice of $x[n]$, $r_{xx}[n-Dk]=\delta[n-Dk]$, giving a clear signal that indicates when $x[n-Dk]$ was received.

Properties of Autocorrelation / Useful TidBits

  1. Hemertian Symmetry: $$r_{xx}[-n]=r_{xx}^*[n]$$ $$r_{xx}[n]=r_{xx}^*[-n]$$ Proof: $$r_{xx}[-n]=x[-n]*x^*[n]=x^*[n]*x[-n]$$ $$r_{xx}^*[n]=x^*[n]*x[-n]$$
  2. Zero property: $$|r_{xx}[n]|\leq |r_{xx}[0]| \forall n$$
  3. DTFT: $$\sum_{k=-\infty}^{\infty}r_{xx}[n]e^{-j\omega n} \geq 0, \in \mathbb{R}, \forall \omega$$
  4. Time Invariance: $$x[n]*x^*[-n]=x[n]*x^*[-n]|_{n=n-n_0}$$
  5. Frequency Shift: $$y[n]=e^{j(\omega_0 n + \theta_0)}x[n]\Rightarrow r_{yy}[n]=e^{j\omega_0 n}r_{xx}[n]$$
  6. LTI Systems: $$y[n] = x[n]*h[n] \Rightarrow r_{yy}[n]=r_{xx}[n] * r_{hh}[n]$$
  7. Geometric Autocorrelation: $$x[n] = a^nu[n] \Rightarrow r_{xx}[n]=\frac{1}{1-a^2}a^{|n|}, |a|<1$$
  8. Autocorrelation in the Frequency Domain (see properties 3 and 4 in the Z-transform properties section): $$r_{xx}[n]=x[n]*x^*[-n]\Rightarrow S_{xx}(e^{j\omega})=X(e^{j\omega})X^*(e^{j\omega})=|X(e^{j\omega})|^2$$