Linear Shift-Invariant Systems

Linear If x(t) and y(t) are two input signals to a system, the system is linear if H[a*x(t) + b*y(t)] = aH[x(t)] + bH[y(t)] Where H specifies the transformation performed on the signal by the system

Shift Invariant If x O (t) = H[x(t)] then H[x(t-τ)] = x O (t - τ)

Properties of an LSI If the input is x(t) = A Cos(2πf 0 t + θ) The response to x(t) is H[x(t)] = A H[Cos(2πf 0 t + θ)] = A out Cos(2πf 0 t + θ out )]

Transfer function of an LSI Consider the response of an LSI to a complex sinusoid

If input is periodic then If input is finite duration

The Fourier Transform of the impulse response function is the transfer function of the linear system. The Inverse Fourier Transform of the transfer function is the impulse response function of the linear system. This is a very powerful result. The easiest way to design a filter is to select an impulse response function

Designing a filter A bandpass filter can be designed by taking an impulse response function that starts at t=0, reaches a single peak and declines to zero with time. The longer the impulse response, the narrower the filter. To set the center frequency,fc, of the filter, multiply it by Cos(2 π fc t).

Predicting the filter’s output

Transfer function with zero phase shift Consider a rectangular filter with no phase shift