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Why do we need DFT
Why do we need the Discrete Fourier Transform (DFT) although we have the Discrete-time Fourier Transform (DTFT)?
The DTFT X(e^jω) of a sequence is a continuous function of ω and needs an infinite sum over n, so it cannot be computed or stored on a computer. The DFT is needed because:
- It is discrete in frequency. It gives N samples of the DTFT at ω_k = 2πk/N, a finite set of numbers that a computer can store and process.
- It is a finite sum. X(k) = Σ (n = 0 to N−1) x(n) e^(−j2πkn/N) for a finite-length x(n); no infinite summation is required.
- Fast algorithms exist (FFT), which cut the computation from N² to (N/2)log₂N operations.
- Practical uses: spectrum analysis, linear filtering by fast convolution (overlap-add/save), correlation, and filter design.
✓ Answer checked by Claude Opus 5.5 (high effort)
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