Fourier Transform Analysis of LTI Systems

Before this you need to relearn the convolution property of Fourier transform.

The convolution theorem states that convolution in time domain corresponds to multiplication in frequency domain and vice versa: LTI Response to a periodic complex exponential we use Fourier series for periodic signal.

Filter is the process of rejecting sinusoids having particularfrequencies or for a range of frequencies is called

filtering and a system that has this characteristic is called a filter.

There are different form of LTI filter:

LTIs can be viewed as Frequency selective filters:

H(kw) >1, Amplify the frequency component

H (kw )< 1, Attenuate the frequency component

H(kw )= 0, Eliminate or block the freq. component The Ideal Filters: completely rejects sinusoidal inputs having frequency in a certain range.

Ideal filters are: lowpass, high pass, bandpass, bandstop

1. Ideal lowpass filter (LPF)

This is when H(w) = 1 will be -BB

2. Ideal highpass filter (HPF)

This is when H(w) = 1 will be w>B and when H(w)=0 the w<B

3. Ideal Bandpass filter (BPF)

This is when H(w) = 1 will be B1<w<B and when H(w)=0 other are all w.

4. Ideal Bandstop filter (BSF)

This is when H(w) = 1 will be other are all w and when H(w)=0 will be B1<w<B.

Examples and Practice Problems
MIT practice problem and solution

Extra Resources
MIT course and video