Question

Digital signals for FFT are assumed:

• A. Periodic
• B. Non-periodic
• C. Random
• D. DC only

Hint:

Always remember: The Fourier Series assumes a continuous periodic signal. The DFT/FFT assumes a sampled sequence that is periodic in both the time domain and frequency domain.

Answer 1

The Correct Option is A

Concept: The Fast Fourier Transform (FFT) is an optimized algorithm used to compute the Discrete Fourier Transform (DFT) of a sampled sequence. The mathematical definition of the DFT inherently operates under the premise that the finite segment of the signal observed within the data window repeats infinitely over time.

Step 1: Examine the mathematical basis of the DFT/FFT.
The Discrete Fourier Transform maps a finite sequence of \( N \) discrete time-domain samples, \( x[n] \), to a finite sequence of \( N \) frequency-domain components, \( X[k] \). The inverse transform is defined as: \[ x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j \frac{2\pi}{N} k n} \] The complex exponential kernel functions, \( e^{j \frac{2\pi}{N} k n} \), are periodic functions with a period of \( N \). Because the transform is built entirely from these periodic complex exponentials, the reconstructed time-domain signal \( x[n] \) is periodic: \[ x[n + N] = x[n] \quad \text{for all integers } n \]

Step 2: Understand the physical consequences of this assumption.
Because the FFT treats the sampled block as one cycle of an infinitely repeating periodic waveform, any mismatch between the starting value and ending value of the sampled block introduces a sharp discontinuity when wrapped around. This boundary jump creates an artifact known as spectral leakage. To minimize this effect, windowing functions are applied to bring the boundaries smoothly to zero, satisfying the underlying assumption of periodicity without sharp edges. Thus, digital signals for FFT are assumed to be periodic.