Notes › ENGR 2341: Signals and Systems Lecture 4
Discrete Time Systems
232 words 1 min Modified
Table of Contents
- A discrete-time system (filter) operates on an input sequence $x[n]$ and produces an output sequence $y[n]$
- Common representations:
- Difference equations
- Impulse response/convolution sum
- Transfer/system function
- Frequency response
System Representations
| Continuous-time | Discrete-time |
|---|---|
| $y(t) = f(y(t), x(t))$ | $y[n] = \sum_{\ell=2}^N a_{\ell}y[n-\ell]+\sum_{k=0}^Mb_{k}x[n-k]$ |
| $y(t) = x(t) * h(t) = \int_{-\infty}^{\infty}x(t-\tau)h(\tau) \ \mathrm{d}\tau$ | $y[n] = h[n] * x[n] = \sum_{k=-\infty}^\infty h[k]x[n-k]$ |
| $H(s), \ H(j\omega)$ | $H(z), \ H(e^{j\hat{\omega}})$ |
Impulse Response
- Impulse responses in discrete-time are labeled $h[n]$
System Properties
- Practically the same as continuous systems (see causality, linearity, time invariance, and BIBO stability)
- For BIBO stability, the system must be absolutely summable
FIR and IIR Filters
- Finite impulse response (FIR) and infinite impulse response (IIR) filters are two specific discrete-time systems
FIR
- Properties:
- LTI
- Always stable
- Ambiguous causality
- Output depends only on the input $x[n]$
- $b_k$ is the $(M+1)$ filter coefficients
- Filter order is $M$
- Response determined with difference equation or convolution sum
IIR
- Properties:
- LTI iff I/O is 0 before first applied input at $n_0$
- $x[n], y[n] = 0, n \lt n_{0}$
- Ambiguous stability
- Ambiguous causality
- LTI iff I/O is 0 before first applied input at $n_0$
- Output depends on $x[n]$ and previous outputs $y[n]$ (recursive filter)
- $b_k$ and $a_\ell$ are the feedforward and feedback coefficients, respectively
- $(N+M+1)$ total coefficients
- Filter order is $\max(N, M)$
- Response determined with the difference equation
References
- Discrete-time systems (course handout)
Sources
- Discrete-time systems