3.10 Frequency response

Recall from [link] that the convolution integral

has the Fourier Transform:

where $H\left(j\Omega \right)$ and $X\left(j\Omega \right)$ are the Fourier Transforms of $h\left(t\right)$ and $x\left(t\right)$ , respectively. Solving for $H\left(j\Omega \right)$ gives the frequency response :

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 $\Omega ={\Omega }_0$ , $\left|H\right(j{\Omega }_0\left)\right|$ gives the factor the frequency component of $x\left(t\right)$ at $\Omega ={\Omega }_0$ would be scaled by. The phase of the frequency response at $\Omega ={\Omega }_0$ , $\angle H\left(j{\Omega }_0\right)$ gives the phase shift the component of $x\left(t\right)$ at $\Omega ={\Omega }_0$ would undergo. This idea will be discussed in greater detail in [link] . 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. [link] shows examples of these various filter types.