Frequency response and the Fourier series
Recall that if the input to an LTI system H is a complex exponential signal e ∈ [Time→ Complex] where for all t ∈ Time,
then the output can be written
where H(ω) is (possibly complex-valued) number that is a property of the system. H(ω) is called the frequency response at frequency ω. It is equal to the output at time zero y(0) when the input is exp(jωt). H itself is a function H: Reals → Complex that in principle can be evaluated for any frequency ω ∈ Reals, including negative frequencies.
Recall further that if an input x(t) to the system H is a periodic signal with period p, then it can (usually) be give as a Fourier series,
By linearity and time invariance, if this is the input, then the output is
Linearity tells us that if the input is decomposed into a sum of components,
then the output can be decomposed into a sum of components where each component
is the response of the system to a single input component. Linearity together
with time invariance tells us that each component, which is a complex exponential,
is simply scaled. Thus, the output is given by a Fourier series with
coefficients X_{k}H(kω_{0}).
This major result tells us:
- There are no frequency components in the output that were not in the input. The output consists of the same frequency components as the input, but with each component individually scaled.
- LTI systems can be used to enhance or suppress certain frequency components. Such operations are called filtering.
- The frequency response function characterizes which frequencies are enhanced or suppressed, and also what phase shifts might be imposed on individual components by the system.