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 xO(t) = H[x(t)] then H[x(t-τ)] = xO(t - τ)

Properties of an LSI

• If the input is x(t) = A Cos(2πf0t + θ)

• The response to x(t) is H[x(t)] =

• A H[Cos(2πf0t + θ)] = Aout Cos(2πf0t + θout)]

Transfer function of an LSI

• Consider the response of an LSI to a complex sinusoid

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fj

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If input is periodic then

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• 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

Figure 5.4. An arbitrary waveform. Thehorizontal line at t0 represents the input of awaveform of this sort to a LSI system, up to andincluding time t0.

Figure 5.4. An arbitrary waveform. Thehorizontal line at t0 represents the input of awaveform of this sort to a LSI system, up to andincluding time t0.

Figure 5.5. A digitized version of the waveformshown in Figure 5.5 is depicted up to andincluding the digitized waveform as it exists attime t0. The waveform has been digitized at arate of 20 kHz.

Figure 5.5. A digitized version of the waveformshown in Figure 5.5 is depicted up to andincluding the digitized waveform as it exists attime t0. The waveform has been digitized at arate of 20 kHz.

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)]([)()]([

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Transfer function with zero phase shift

• Consider a rectangular filter with no phase shift

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