The Behavior of Fourier Transform for Periodic Digital Signals: A Comprehensive Analysis

Understanding the behavior of the Fourier Transform for periodic digital signals is crucial in many applications in signal processing and analysis. Unlike the Fourier Transform of non-periodic signals, which results in continuous frequency components, the Fourier Transform (or Fourier Series) of a periodic digital signal leads to a unique set of frequency components. In this article, we will delve into the significant properties and behaviors of the Fourier Transform for periodic digital signals, particularly focusing on the often-asked question: do periodic digital signals always decrease in the frequency domain?

Periodic Signals and Fourier Series

A periodic digital signal, unlike a non-periodic signal, can be perfectly represented by a Fourier Series rather than a Fourier Transform. A Fourier Series decomposes a periodic signal into a sum of sinusoids and complex exponentials at discrete, evenly-spaced frequencies. This representation is incredibly useful in signal analysis as it allows us to understand the signal in a different domain—namely, the frequency domain.

Frequency Spectrum of Periodic Digital Signals

The frequency spectrum of a periodic digital signal consists of discrete frequency components at integer multiples of the fundamental frequency. These components are often referred to as harmonics. The magnitude of these frequency components can vary widely depending on the specific characteristics of the signal. Some frequencies may have significantly higher amplitudes than others, leading to a non-monotonic (non-decreasing or non-increasing) frequency spectrum.

Magnitude Spectrum and Signal Characteristics

The magnitude spectrum of a periodic signal, which is derived from the Fourier Series coefficients or the Discrete Fourier Transform (DFT) coefficients for sampled signals, can exhibit various behaviors. In some cases, the magnitude spectrum of a periodic signal may not decrease as frequency increases. Instead, it can have peaks and troughs, indicating the presence of specific frequency components that dominate the signal. This behavior is highly dependent on the specific characteristics of the signal and its underlying structure.

Real-World Constraints on Frequency Spectrum

In the real world, all real waveforms cannot have a frequency spectrum that decreases indefinitely and must eventually taper off. This is because infinite frequency and zero rise time are physically impossible. As a result, the frequency spectrum of any real-world periodic signal will eventually stabilize or start to decrease as the frequency approaches infinity.

Sampled Data Time Signal vs. Binary Periodic Signal

A sampled data time signal, which is a representation of a time-domain signal at discrete intervals, has a periodic spectrum when considered over its period. In contrast, a binary periodic signal, which consists of only two levels (e.g., 0 and 1), will have a periodic spectrum that repeats itself every period. The spectrum of a periodic signal cannot be monotonically decreasing if it keeps bumping into copies of itself, given the periodic nature of the signal.

Nyquist Interval and Spectral Envelope

When considering the spectrum within a single Nyquist interval, the behavior of the spectral envelope can be analyzed. Within a single Nyquist interval, the periodic copies of the spectrum can indeed be monotonically decreasing, provided the time sequence does not contain impulses. However, if the signal includes impulses, such as those found at the boundaries of a Finite Impulse Response (FIR) filter designed by the Remez exchange algorithm or other Chebyshev approximation methods, the spectral envelope may contain a constant level of spectral sidelobes. This is due to the sharp transitions or impulses in the time domain being represented as smaller, oscillatory components in the frequency domain.

Understanding the behavior of the Fourier Transform for periodic digital signals is essential for proper signal analysis in a variety of applications. From telecommunications to audio engineering, the properties of periodic signals and their frequency spectra play a crucial role. By recognizing that the Fourier Transform or series of a periodic digital signal is not always monotonically decreasing, we can better analyze and interpret complex signals.