1 / 15

Linear Shift-Invariant Systems

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.

inari
Download Presentation

Linear Shift-Invariant Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Linear Shift-Invariant Systems

  2. 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

  3. Shift Invariant • If xO(t) = H[x(t)] then H[x(t-τ)] = xO(t - τ)

  4. 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)] Properties of an LSI

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

  6. If input is periodic then If input is finite duration

  7. 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

  8. 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).

  9. Predicting the filter’s output

  10. Transfer function with zero phase shift • Consider a rectangular filter with no phase shift

More Related