Math Problem Statement

Compute the sequence x(n) for which the DFT is { 20, -4+4j, -4, -4-4j}

Solution

To compute the sequence x(n)x(n) for which the Discrete Fourier Transform (DFT) is given as {20,4+4j,4,44j}\{ 20, -4 + 4j, -4, -4 - 4j \}, we need to use the inverse DFT (IDFT) formula.

The IDFT is defined as:

x(n)=1Nk=0N1X(k)ej2πNknfor n=0,1,2,,N1,x(n) = \frac{1}{N} \sum_{k=0}^{N-1} X(k) e^{j\frac{2\pi}{N}kn} \quad \text{for } n = 0, 1, 2, \dots, N-1,

where:

  • X(k)X(k) is the DFT of the sequence.
  • NN is the length of the sequence (in this case, N=4N = 4).
  • jj is the imaginary unit.

Given X(k)={20,4+4j,4,44j}X(k) = \{ 20, -4 + 4j, -4, -4 - 4j \}, let's compute x(n)x(n) for n=0,1,2,3n = 0, 1, 2, 3.

The IDFT formula for N=4N = 4 becomes:

x(n)=14k=03X(k)ej2π4kn.x(n) = \frac{1}{4} \sum_{k=0}^{3} X(k) e^{j\frac{2\pi}{4}kn}.

Let's calculate each value of x(n)x(n) step-by-step.The sequence x(n)x(n) for which the DFT is {20,4+4j,4,44j}\{ 20, -4 + 4j, -4, -4 - 4j \} is:

x(n)={2,4,6,8}.x(n) = \{ 2, 4, 6, 8 \}.

Would you like more details on the process or have any questions?

Here are 5 related questions you might find interesting:

  1. How is the DFT related to the Fourier series for periodic signals?
  2. What is the significance of the imaginary unit jj in the DFT?
  3. How would you compute the DFT for a sequence longer than 4 points?
  4. What is the effect of zero-padding a sequence before computing the DFT?
  5. How does the DFT differ from the continuous Fourier transform?

Tip: The DFT assumes periodicity in the input sequence, which means the signal is treated as if it repeats indefinitely.

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Math Problem Analysis

Mathematical Concepts

Discrete Fourier Transform (DFT)
Inverse Discrete Fourier Transform (IDFT)
Complex numbers

Formulas

IDFT formula: x(n) = (1/N) * sum(X(k) * e^(j*2*pi*k*n/N))

Theorems

-

Suitable Grade Level

Advanced