Notes › ENGR 2341: Signals and Systems Lecture 16

Discrete-time Frequency Response

calendar_today March 27, 2025 article 154 words access_time 1 min replay Modified April 1, 2025

Table of Contents

Discrete-time Systems

Represented by
- Difference equations
- Impulse response (convolution sum)
- Frequency response
- Transfer (system) function
$h[n]$, the impulse response, is the system response to a unit impulse input

$$h[n] = y[n] \vert_{x[n]=\delta[n]}$$

BIBO stable if $h[n]$ is absolutely summable

$$y[n] = h[n] * x[n] = \sum_{k=-\infty}^{\infty}h[k]x[n-k]$$

Frequency Response

Describes what the system does to sinusoidal inputs
- Commonly broken up into magnitude and phase
Frequency response of a system can be found by taking the DTFT of $h[n]$

$$H(e^{j\hat{\omega}}) = \sum_{n=-\infty}^\infty h[n]e^{-j\hat{\omega}n} = |H(e^{j\hat{\omega}})|\angle H(e^{j\hat{\omega}})$$

Sinusoidal Response

Response to sinusoidal inputs is a sinusoid of same frequency with an amplitude change and phase shift dependent on $H(e^{j\hat{\omega}})$
The sinusoidal input must be two-sided and the frequencies of the sinusoidal components cannot change

$$x[n] = A_{0} + \sum_{k=1}^{N}A_{k}\cos(\hat{\omega_{k}}n+\phi_{k}), -\infty \lt n \lt \infty$$$$ \begin{align*} y[n] &= H(e^{j_{0}})A_{0} + \sum_{k=1}^{N}H(e^{j \hat{\omega_{k}}})A_{k}\cos(\hat{\omega_{k}}n + \phi_{k}) \\ &= \left|H(e^{j_{0}})\right|A_{0} + \sum_{k=1}^N \left|H(e^{j\hat{\omega_{k}}})\right|A_{k}\cos\left(\hat{\omega_{k}}n+\phi_{k}+\angle H(e^{j \hat{\omega_{k}}})\right) \end{align*} $$

References

Dt frequency response (course handout)

Sources

Dt frequency response

Graph

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