Signals and Systems/Frequency Response

Frequency Response
Systems respond differently to inputs of different frequencies. Some systems may amplify components of certain frequencies, and attenuate components of other frequencies. The way that the system output is related to the system input for different frequencies is called the frequency response of the system.

The frequency response is the relationship between the system input and output in the Fourier Domain.



In this system, X(j&omega;) is the system input, Y(j&omega;) is the system output, and H(j&omega;) is the frequency response. We can define the relationship between these functions as:


 * $$Y(j\omega) = H(j\omega)X(j\omega)$$


 * $$\frac{Y(j\omega)}{X(j\omega)} = H(j\omega)$$

The Frequency Response Functions
Since the frequency response is a complex function, we can convert it to polar notation in the complex plane. This will give us a magnitude and an angle. We call the angle the phase.

Amplitude Response
For each frequency, the magnitude represents the system's tendency to amplify or attenuate the input signal.


 * $$A\left( \omega \right) = \left| {H\left( {j\omega } \right)} \right|$$

Phase Response
The phase represents the system's tendency to modify the phase of the input sinusoids.
 * $$\phi \left( \omega \right) = \angle H\left( {j\omega } \right)$$.

The phase response, or its derivative the group delay, tells us how the system delays the input signal as a function of frequency.

Filters
An important concept to take away from these examples is that by desiging a proper system called a filter, we can selectively attenuate or amplify certain frequency ranges. This means that we can minimize certain unwanted frequency components (such as noise or competing data signals), and maximize our own data signal

We can define a "received signal" r as a combination of a data signal d and unwanted components v:


 * $$r(t) = d(t) + v(t)$$

We can take the energy spectral density of r to determine the frequency ranges of our data signal d. We can design a filter that will attempt to amplify these frequency ranges, and attenuate the frequency ranges of v. We will discuss this problem and filters in general in the next few chapters. More advanced discussions of this topic will be in the book on Signal Processing.