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Frequency Response

Module by: Carlos E. Davila. E-mail the author

Recall from (Reference) that the convolution integral

y ( t ) = ∫ - ∞ ∞ x ( τ ) h ( t - τ ) d τ y ( t ) = ∫ - ∞ ∞ x ( τ ) h ( t - τ ) d τ (1)

has the Fourier Transform:

Y ( j Ω ) = H ( j Ω ) X ( j Ω ) Y ( j Ω ) = H ( j Ω ) X ( j Ω ) (2)

where H(jΩ)H(jΩ) and X(jΩ)X(jΩ) are the Fourier Transforms of h(t)h(t) and x(t)x(t), respectively. Solving for H(jΩ)H(jΩ) gives the frequency response:

H ( j Ω ) = Y ( j Ω ) X ( j Ω ) H ( j Ω ) = Y ( j Ω ) X ( j Ω ) (3)

The frequency response, the Fourier Transform of the impulse response of a filter, is useful since it gives a highly descriptive representation of the properties of the filter. The frequency response can be considered to be the gain of the filter, expressed as a function of frequency. The magnitude of the frequency response evaluated at Ω=Ω0Ω=Ω0, |H(jΩ0)||H(jΩ0)| gives the factor the frequency component of x(t)x(t) at Ω=Ω0Ω=Ω0 would be scaled by. The phase of the frequency response at Ω=Ω0Ω=Ω0, ∠H(jΩ0)∠H(jΩ0) gives the phase shift the component of x(t)x(t) at Ω=Ω0Ω=Ω0 would undergo. This idea will be discussed in greater detail in (Reference). A lowpass filter is a filter which only passes low frequencies, while attenuating or filtering out higher frequencies. A highpass filter would do just the opposite, it would filter out low frequencies and allow high frequencies to pass. Figure 1 shows examples of these various filter types.

Figure 1: Different filter types: (a) lowpass, (b) bandpass, (c) highpass.
Figure 1 (ch3_filter_types.png)

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