EECS20N: Signals and Systems

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,

e(t) = exp(jωt) = cos(ωt) + j sin(ωt).

then the output can be written

y( t) = H(ω) exp(jωt)

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 XkH(kω0).
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.