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By means of the DFT and IDFT, determine the response of the FIR filter with impulse response h(n)={1,2,3} to the input sequence x(n)={1,2,2,1}?
- a){1,4,11,9,8,3}
- b){1,4,9,11,8,3}
- c){1,4,9,11,3,8}
- d){1,4,9,3,8,11}
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
By means of the DFT and IDFT, determine the response of the FIR filter...
Explanation: The input sequence has a length N=4 and impulse response has a length M=3. So, the response must have a length of 6(4+3-1).
We know that, Y(k)=X(k).H(k)
Thus we obtain Y(k)={36,-14.07-j17.48,j4,0.07+j0.515,0,0.07-j0.515,-j4,-14.07+j17.48}
By applying IDFT to the above sequence, we get y(n)={1,4,9,11,8,3,0,0}
Thus the output of the system is {1,4,9,11,8,3}.
We know that, Y(k)=X(k).H(k)
Thus we obtain Y(k)={36,-14.07-j17.48,j4,0.07+j0.515,0,0.07-j0.515,-j4,-14.07+j17.48}
By applying IDFT to the above sequence, we get y(n)={1,4,9,11,8,3,0,0}
Thus the output of the system is {1,4,9,11,8,3}.
Most Upvoted Answer
By means of the DFT and IDFT, determine the response of the FIR filter...
Given:
Impulse response of FIR filter, h(n) = {1, 2, 3}
Input sequence, x(n) = {1, 2, 2, 1}
To find:
Response of the FIR filter to the input sequence using DFT and IDFT
Solution:
The response of the FIR filter can be determined by convolving the impulse response of the filter with the input sequence. This can be done in the frequency domain using the Discrete Fourier Transform (DFT) and Inverse Discrete Fourier Transform (IDFT).
Step 1: Taking DFT of the impulse response and input sequence
1. Compute the DFT of the impulse response h(n) and the input sequence x(n) using the formula:
X(k) = Σ[x(n) * exp(-j*2π*k*n/N)]
H(k) = Σ[h(n) * exp(-j*2π*k*n/N)]
where, N is the length of the sequence (in this case, N = 4).
DFT of h(n):
H(0) = 1 * exp(-j*2π*0*0/4) + 2 * exp(-j*2π*0*1/4) + 3 * exp(-j*2π*0*2/4) = 1 + 2 + 3 = 6
H(1) = 1 * exp(-j*2π*1*0/4) + 2 * exp(-j*2π*1*1/4) + 3 * exp(-j*2π*1*2/4) = 1 - 2 + 3 = 2
H(2) = 1 * exp(-j*2π*2*0/4) + 2 * exp(-j*2π*2*1/4) + 3 * exp(-j*2π*2*2/4) = 1 + 2 - 3 = 0
H(3) = 1 * exp(-j*2π*3*0/4) + 2 * exp(-j*2π*3*1/4) + 3 * exp(-j*2π*3*2/4) = 1 - 2 - 3 = -4
DFT of x(n):
X(0) = 1 * exp(-j*2π*0*0/4) + 2 * exp(-j*2π*0*1/4) + 2 * exp(-j*2π*0*2/4) + 1 * exp(-j*2π*0*3/4) = 1 + 2 + 2 + 1 = 6
X(1) = 1 * exp(-j*2π*1*0/4) + 2 * exp(-j*2π*1*1/4) + 2 * exp(-j*2π*1*2/4) + 1 * exp(-j*2π*1*3/4) = 1 - 2 + 2 - 1 = 0
X
Impulse response of FIR filter, h(n) = {1, 2, 3}
Input sequence, x(n) = {1, 2, 2, 1}
To find:
Response of the FIR filter to the input sequence using DFT and IDFT
Solution:
The response of the FIR filter can be determined by convolving the impulse response of the filter with the input sequence. This can be done in the frequency domain using the Discrete Fourier Transform (DFT) and Inverse Discrete Fourier Transform (IDFT).
Step 1: Taking DFT of the impulse response and input sequence
1. Compute the DFT of the impulse response h(n) and the input sequence x(n) using the formula:
X(k) = Σ[x(n) * exp(-j*2π*k*n/N)]
H(k) = Σ[h(n) * exp(-j*2π*k*n/N)]
where, N is the length of the sequence (in this case, N = 4).
DFT of h(n):
H(0) = 1 * exp(-j*2π*0*0/4) + 2 * exp(-j*2π*0*1/4) + 3 * exp(-j*2π*0*2/4) = 1 + 2 + 3 = 6
H(1) = 1 * exp(-j*2π*1*0/4) + 2 * exp(-j*2π*1*1/4) + 3 * exp(-j*2π*1*2/4) = 1 - 2 + 3 = 2
H(2) = 1 * exp(-j*2π*2*0/4) + 2 * exp(-j*2π*2*1/4) + 3 * exp(-j*2π*2*2/4) = 1 + 2 - 3 = 0
H(3) = 1 * exp(-j*2π*3*0/4) + 2 * exp(-j*2π*3*1/4) + 3 * exp(-j*2π*3*2/4) = 1 - 2 - 3 = -4
DFT of x(n):
X(0) = 1 * exp(-j*2π*0*0/4) + 2 * exp(-j*2π*0*1/4) + 2 * exp(-j*2π*0*2/4) + 1 * exp(-j*2π*0*3/4) = 1 + 2 + 2 + 1 = 6
X(1) = 1 * exp(-j*2π*1*0/4) + 2 * exp(-j*2π*1*1/4) + 2 * exp(-j*2π*1*2/4) + 1 * exp(-j*2π*1*3/4) = 1 - 2 + 2 - 1 = 0
X
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By means of the DFT and IDFT, determine the response of the FIR filter with impulse response h(n)={1,2,3} to the input sequence x(n)={1,2,2,1}?a){1,4,11,9,8,3}b){1,4,9,11,8,3}c){1,4,9,11,3,8}d){1,4,9,3,8,11}Correct answer is option 'B'. Can you explain this answer?
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By means of the DFT and IDFT, determine the response of the FIR filter with impulse response h(n)={1,2,3} to the input sequence x(n)={1,2,2,1}?a){1,4,11,9,8,3}b){1,4,9,11,8,3}c){1,4,9,11,3,8}d){1,4,9,3,8,11}Correct answer is option 'B'. Can you explain this answer? for Electrical Engineering (EE) 2024 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about By means of the DFT and IDFT, determine the response of the FIR filter with impulse response h(n)={1,2,3} to the input sequence x(n)={1,2,2,1}?a){1,4,11,9,8,3}b){1,4,9,11,8,3}c){1,4,9,11,3,8}d){1,4,9,3,8,11}Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Electrical Engineering (EE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for By means of the DFT and IDFT, determine the response of the FIR filter with impulse response h(n)={1,2,3} to the input sequence x(n)={1,2,2,1}?a){1,4,11,9,8,3}b){1,4,9,11,8,3}c){1,4,9,11,3,8}d){1,4,9,3,8,11}Correct answer is option 'B'. Can you explain this answer?.
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