Lecture 11 - Notes

January 27, 2016

Difference Equations

definition: The Difference Equation is a formula for computing an output sample at time $n$ based on past and present input samples and past output samples in the time domain (source). We may write the general, causal, LTI difference equation as follows:

$$\begin{align} a_0 y[n] + a_1 y[n - 1] + ... + a_N y[n - N] = b_0 x[n] &+ b_1 x[n-1] + ... + b_M x[n - M] \newline \end{align}$$

isolating $y[n]$ (and scaling the coefficients by $\frac{1}{a_0}$),

$$\begin{align} y[n] = b_0 x[n] &+ b_1 x[n-1] + ... + b_M x[n - M] \newline &- a_1 y[n - 1] - ... - a_N y[n - N] \newline \end{align}$$

or, more concisely,

$$\begin{align} y[n] &= \sum_{i=0}^{M} b_i x[n - i] - \sum_{j=1}^{N} a_j y[n-j] \end{align}$$

where $x$ is the input signal, $y$ is the output signal and $a_j, b_i, i, j$ are the constant coefficients. Alternatively, another way of writing the Difference Equation is,

$$\begin{align} \sum_{i=0}^{M} b_i x[n - i] = \sum_{j=0}^{N} a_j y[n-j] \end{align}$$

note the change of the bound $j$ to accommodate the inclusion of $y[n]$.

Example

Consider,

$$\begin{align} y[n] – ay[n-1]=x[n] \end{align}$$

What is the impulse response of the system?

This system is causal, therefore the initial rest condition applies, and

$$\begin{align} h[n] = 0 \text{ for } n \lt 0 \end{align}$$

So, if $x[n] = \delta[n]$,

$$\begin{align} h[n] &= \delta[n] + a h[n-1] \\\\ h[0] &= \delta[0] + a h[-1] = 1 \\\\ h[1] &= \delta[1] + a h[0] = a \\\\ h[2] &= \delta[2] + a h[1] = a^2 \\\\ h[3] &= \delta[3] + a h[2] = a^3 \\\\ &~~\vdots \\\\ h[n] &= a^n \underbrace{u[n]}_\text{Causal} \\\\ \end{align}$$

For what ranges of $a$ will the system be stable?

We note that this absolutely summable for $|a| \lt 1$. This means the system is stable for $|a| \lt 1$.

Condition of Initial Rest

definition: We need addition conditions along with the difference equation. The most common is initial rest, where an input $x[n] = 0$ for all $n \lt n_0$ leads to $y[n] = 0$ for all $n \lt n_0$.

Non-recursive Equations

Given,

$$\begin{align} y[n] &= \sum_{i=0}^{M} b_i x[n - i] - \sum_{j=1}^{N} a_j y[n-j] \end{align}$$

if $N = 0$, then,

$$\begin{align} y[n] &= \sum_{i=0}^{M} b_i x[n - i] \end{align}$$

and,

$$\begin{align} h[n] &= \sum_{i=0}^{M} b_i \delta[n - i] \end{align}$$

Eigenfunctions

Recall

Vector $x$ is an eigenvector for matrix $A$ if,

$$\begin{align} Ax = \lambda x \end{align}$$

where $\lambda$ is the eigenvalue.

Eigenfunctions

Can we extend the concept of eigenvectors to eigenfunctions of a system?

Given a system $x[n]\xrightarrow{T} y[n]$ we can write,

$$\begin{align} y[n] = T\left\{ x \left[n\right] \right\} = \lambda x \end{align}$$

Example

Given a moving average system,

$$\begin{align} y[n] = \frac{x[n] + x[n-1]}{2} \end{align}$$

suppose $x[n] = \delta[n]$,

$$\begin{align} y[n] &= \frac{\delta[n] + \delta[n-1]}{2} \\\\ &\neq \lambda x \end{align}$$

which is not an eigenfunction. Let's try again with $x[n] = e^{j \omega_0 n}$, then,

$$\begin{align} y[n] &= \frac{e^{j \omega_0 n} + e^{j \omega_0 (n - 1)}}{2} \\\\ &= e^{j \omega_0 n} \cdot \frac{1 + e^{-j \omega_0}}{2} \\\\ &= \underbrace{\left(\frac{1 + e^{-j \omega_0}}{2}\right)}_\lambda \underbrace{e^{j \omega_0 n}}_x\\\\ &= \lambda x\\\\ \end{align}$$

Why Eigenfunctions?

If the input to an LTI system is represented as a linear combination of complex exponentials, then the output can also be represented as a linear combination of the same complex exponential signals.

The Eigenfunction Property

definition: The eigenfunction property states given

$$x[n] = e^{j \omega_0 n} \to y[n] = H \left(e^{j \omega_0} \right) \cdot e^{j \omega_0 n}$$

if $x[n] = \sum_k \alpha_k e^{j \omega_k n}$, then,

$$\begin{align} y[n] = \sum_k \alpha_k H \left(e^{j \omega_k} \right) e^{j \omega_k n} \end{align}$$