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# Convolution and filtering the convolution theorem

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Convolution and Filtering: The Convolution Theorem Applications of the Convolution Theorem Example 1: Frequency Analysis of Convolution If the impulse response is a Gaussian with standard deviation σ , what does this do to frequencies going through the system? -10 -5 5 10 0.1 0.2 0.3 0.4 0.5 0.6 20 40 60 80 100 120 0.1 0.2 0.3 0.4 Time Domain Frequency Domain Impulse Response Transfer Function

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Convolution and Filtering: The Convolution Theorem Applications of the Convolution Theorem Example 2: Truncating a Signal What is the Fourier Transform of a truncated signal? f ( t ) = 10 cos ( 2 π st ) + 10 if - π t π 0 otherwise -4 -2 2 4 5 10 15 20 ( 10 cos ( 2 π st ) + 10 )( Rect 2 π ( t ))
Convolution and Filtering: The Convolution Theorem Applications of the Convolution Theorem Example 2 (cont’d) -4 -2 2 4 5 10 15 20 50 100 150 200 250 25 50 75 100 125 150 175 200 Time Domain Frequency Domain

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Convolution and Filtering: The Convolution Theorem Applications of the Convolution Theorem Example 3: Frequency Multiplexing Suppose that we modulate (multiply) a carrier signal of a particular frequency s by a signal f ( t ) : f ( t ) cos ( 2 π st ) Fourier Transform: F ( f ( t ) cos ( 2 π st )) = F ( f ( t )) * F ( cos ( 2 π st )) = F ( u ) * [ 1 2 δ ( u - s ) + 1 2 δ ( u + s )] = 1 2 F ( u - s ) + 1 2 F ( u + s )] This just shifts the signal in the frequency domain by the frequency s of the carrier (splits between s and - s ).
Convolution and Filtering: The Convolution Theorem Applications of the Convolution Theorem

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