EE5410 The convolution property
1. if a periodic signal is represented in a Fourier series -- ie., as a linear combination of harmonically related complex exponentials, -- then th response of an LTI system to this input can also be represented by a Fourier series.
2. Because complex exponentials are eigenfunctions of LTI systems, the Fourier series coefficients of the output are those of the input MULTIPLIED by the frequency response of the system evaluated at the corresponding harmonic frequencies.
3. The Fourier transform synthesis equation as an expression for x(t) as a linear combination of complex exponentials.
4. The frequency response H(jw) as the Fourier transform of the system impulse response.
5. The Fourier transform of the impulse response (evaluated w= kw0) is the complex scaling factor that the LTI system applies to the eigenfunction e^(j^kw0t).
6. The Fourier transform maps the convolution of 2 signals into PRODUCT of their Fourier transforms.
Y(jw) = H(jw)X(jw)
7. H(jw), the Fourier transform of the impulse response, is the frequency response, and captures the CHANGE in complex amplitude of the Fourier transform of the input at EACH frequency w.
8. eg., in frequency-selective filtering, we may want to have H(jw) approx. = 1 over one range of frequencies, so that the frequency components in this band experience little or NO attenuation or change due to the system,
while over another range of frequencies we may want to have H(jw0) approx. =0, so that components in this range are eliminated or significantly attenuated.