Lecture 7 - Notes

January 19, 2016

Ways to Describe LTI Systems

1. The Impulse Response
2. Difference Equations
3. Frequency Response
4. Transfer Function

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 $y[n]$.