EECS20N: Signals and Systems

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Week 12

• Freq. Response
• Continuous time
• Moving Average
• Signal Spaces
• Transforms
• Symmetry
• Inverse
• Examples
• Linearity
• Using Linearity
• Constants
• Exponentials
• Sinusoid
• Discrete
• Fourier Transforms
• Convolution
• Signal products

Relationship to Convolution

Suppose an LTI system has impulse response h(n) and frequency response H(ω ). We have seen that if the input to this system is e^{jω n}, then the output is H(ω )e^{jω n}. Suppose the input is instead a signal x with DTFT X. Using the inverse DTFT relation, we know that for all n, .

View this as a summation of exponentials, each with weight X(ω ). An integral, after all, is summation over a continuum. Each term in the summation is X(ω )e^{jω n}. If this term were an input by itself, then the output would be H(ω )X(ω )e^{jω n}. Thus, by linearity, if the input is x, the output should be

.

Comparing to the inverse DTFT relation for y(n), we see that

Y(ω ) = H(ω )X(ω ).

This is the frequency-domain version of convolution

y(n) = (h * x)(n). Exercise: Show that if two discrete-time systems with frequency responses H₁(ω ) and H₂(ω ) are connected in cascade, that the DTFT of the output is given by Y(ω ) = H₁(ω )H₂(ω )X(ω ), where X(ω ) is the DTFT of the input.