Response of the filter to the complex sinusoid is

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(response) of the filter to the complex sinusoid is given by -∞ = - -∞ = = - = k k n j j k k k e Ae b k n x b n y ) ( ˆ ] [ ] [ ϖ φ n j j k k j k e Ae e b ϖ φ ϖ ˆ ˆ = -∞ = - ) ˆ ( ˆ ) ( φ ϖ ϖ + = n j j Ae e H where -∞ = - = k k j k j e b e H ϖ ϖ ˆ ˆ ) ( Response of FIR filter to complex sinusoid Frequency response to sum of sinusoids: The frequency response of an LTI system to a sum of sinusoids can be obtained by making use of the linearity property of the system and the Digital Filter D ) ( ˆ ˆ ) ( ϖ φ ϖ j e H n j A e H + = ) ˆ ( ] [ φ ϖ + = n j Ae n x ) ˆ ( ˆ ) ( ] [ φ ϖ ϖ + = n j j Ae e H n y
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response to complex sinusoid. Let us look at the response to a single sinusoid first. Let ) ˆ cos( ] [ 1 1 1 φ ϖ + = n A n x . The input can be decomposed into a sum of two complex sinusoids by using Euler’s equation: ) ˆ ( 1 ) ˆ ( 1 1 1 1 1 2 2 ] [ φ ϖ φ ϖ + - + + = n j n j e A e A n x Using the linearity of FIR filter, the output of FIR to ] [ n x is ) ˆ ( ˆ 1 ) ˆ ( ˆ 1 1 1 1 1 1 1 ) ( 2 ) ( 2 ] [ φ ϖ ϖ φ ϖ ϖ + - - + + = n j j n j j e e H A e e H A n y ) ˆ ( ) ˆ ( ˆ 1 ) ˆ ( ) ˆ ( ˆ 1 1 1 1 1 1 1 1 1 ) ( 2 ) ( 2 φ ϖ ϖ ϖ φ ϖ ϖ ϖ + - - - + + = n j H j j n j H j j e e e H A e e e H A Using the conjugate symmetry of the frequency response, the output can be expressed as ) ( ) ( 2 ] [ )] ( ˆ [ )] ( ˆ [ ˆ 1 1 ˆ 1 1 1 ˆ 1 1 1 ϖ ϖ φ ϖ φ ϖ ϖ j j e H n j e H n j j e e e H A n y + + - + + + = )) ( ˆ cos( ) ( ] [ 1 1 ˆ 1 1 ˆ 1 ϖ ϖ φ ϖ j j e H n e H A n y + + = Similarly, the response of the LTI system to ) ˆ sin( ] [ 1 1 1 φ ϖ + = n B n x is given by
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)) ( ˆ sin( ) ( ] [ 1 1 ˆ 1 1 ˆ 1 ϖ ϖ φ ϖ j j e H n e H B n y + + = . This result shows a technique to measure the frequency response of an unknown discrete-time system. We can input a sinusoidal signal of frequency ω to the system and measure the output amplitude and the phase shift. The magnitude of the frequency response of the system at the frequency ω is the ratio of the output amplitude to the input amplitude, and the phase of the frequency response is the phase shift. The result can be extended to find the response of the LTI system with the frequency response ) ( ˆ ϖ j e H to = + + = K k k k k n A A n x 1 0 ) ˆ cos( ] [ φ ϖ . The output is expressed as )) ( ˆ cos( ) ( ) ( ] [ ˆ 1 ˆ 0 0 k k j k k k K k j j e H n A e H A e H n y ϖ ϖ φ ϖ + + + = = Example: For the FIR filter with coefficients {1, 2, 1}, find the response when the input is ) 21 20 cos( 3 ) 2 3 cos( 3 4 ] [ n n n x π π π + - + =
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The frequency response of the FIR filter is ) ˆ cos 2 2 ( ) ( ˆ ˆ ϖ ϖ ϖ + = - j j e e H The input has frequencies at 0, 3 / π , and 21 / 20 π . The frequency responses at the input frequencies are 4 2 2 ) ( 0 = + = j e H ; 3 / 3 / 3 / 3 ) 1 2 ( ) ( π π π j j j e e e H - - = + = 21 / 20 21 / 20 21 20 21 / 20 0223 . 0 ) 9777 . 1 2 ( )) cos( 2 2 ( ) ( 21 20 π π π π π j j j j e e e e H - - - = - = + = The response is + + - + = )] ( 2 3 cos[ 3 | ) ( | 4 ) ( ] [ 3 3 0 π π π π j j j e H n e H e H n y ) ( 21 20 cos( 3 | ) ( | 21 20 21 20 π π π j j e H n e H + ) 21 20 21 20 cos( 0669 . 0 ) 6 5 3 cos( 9 16 π π π π - + - + = n n The response of the LTI system to sinusoids obtained from the frequency response is a steady state solution; it means the input has passed to the system for quite a long time.
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