Math Problem Statement
Compute the sequence x(n) for which the DFT is { 20, -4+4j, -4, -4-4j}
Solution
To compute the sequence for which the Discrete Fourier Transform (DFT) is given as , we need to use the inverse DFT (IDFT) formula.
The IDFT is defined as:
where:
- is the DFT of the sequence.
- is the length of the sequence (in this case, ).
- is the imaginary unit.
Given , let's compute for .
The IDFT formula for becomes:
Let's calculate each value of step-by-step.The sequence for which the DFT is is:
Would you like more details on the process or have any questions?
Here are 5 related questions you might find interesting:
- How is the DFT related to the Fourier series for periodic signals?
- What is the significance of the imaginary unit in the DFT?
- How would you compute the DFT for a sequence longer than 4 points?
- What is the effect of zero-padding a sequence before computing the DFT?
- 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
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