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Frequency Response

The Basics

Equation
  • The Frequency Response Characteristic is the Impulse Response of a LTI system to a Complex Sinusoid
  • Also known as the DTFT of \(h[n]\) and the Frequency Response Characteristic
    \(\(\LARGE H(e^{j2\pi f})=\sum\limits_{k=-\infty}^{\infty}h[k]e^{-j2\pi fk}\)\)
  • \(H(e^{j2\pi f})\) exists for every BIBO Stable LTI system as well as some that are not stable
  • Is similar to [[Transfer Function]]:
    \(\(\large y[n]=H(e^{j2\pi f})Xe^{j2\pi fn}\)\)
    \(\(\large Xe^{j2\pi f}\xrightarrow{h[n]}Ye^{j2\pi f}\)\)
Derivation
  1. Definition of Impulse Response and Convolution \(\(\large y[n]=x[n]\ast h[n]=\sum\limits_{k=-\infty}^{\infty}x[n-k]h[k]\)\)
  2. Replace input with a Complex Sinusoid \(\(\large =\sum\limits_{k=-\infty}^{\infty}Xe^{j2\pi f(n-k)}h[k]\)\)
  3. Pull out piece which does not vary with k \(\(\large =Xe^{j2\pi f(n)}\bigg(\sum\limits_{k=-\infty}^{\infty}e^{-j2\pi fk}h[k]\bigg)\)\)
  4. Rewrite terms as freq. resp. \(\(\LARGE =Xe^{j2\pi fn}H(e^{j2\pi f})\)\)
  5. The response of an LTI system to a sinusoidal input will be a sinusoid with the same frequency
    • Because both the input and the response have the same frequency, the output will too!
Periodicity
  • \(H(e^{j2\pi f})\) is periodic with a period of 1:
    • \(H(e^{j2\pi (f+1)})=H(e^{j2\pi f})\)
    • (Period of sine and cosine is 2pi, so sin(2pif) has a period of 1)
  • See Normalized Frequency
Examples

Example Calculation using DTFT of Impulse Response

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Example Calculation using definition of Complex Sinusoid

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