Complex-valued exponential sequence

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In digital signal processing, we utilize various elementary sequences for the purpose of analysis. In this series, we will see such sequences. One such elementary sequence is the real-valued exponential sequence. (see the articles on unit sample sequence, unit step sequence, real-valued exponential sequence)

A complex-valued exponential sequence in signals and systems is a discrete-time sequence that exhibits complex exponential behavior. It is characterized by complex numbers raised to the power of the index. The general form of a complex-valued exponential sequence is given by:

\[x[n] = e^{ \alpha n }e^{ j \omega n } = e^{\left( \alpha + j \omega \right) n } = cos \left[ \left( \alpha + j \omega \right) n \right] + j sin \left[ \left( \alpha + j \omega \right) n \right], \; \forall n \]

where:

  • x[n] is the value of the sequence at index n.
  • \(\alpha\) acts as an attenuation factor if \(\alpha \lt 0\) or as an amplification factor for \(\alpha \gt 0\)
  • \(j\) is the indeterminate satisfying \(j^2 = -1\) (imaginary unit).
  • \(\omega\) is the angular frequency in radians per sample.

The complex nature is indicated by the presence of the indeterminate \(j\) in the exponent.

The python function to generate a complex exponential function is given below

import numpy as np
import matplotlib.pyplot as plt

def complex_exponential_sequence(n, alpha, omega):
    return  np.exp((alpha + 1j * omega) * n)

n = np.linspace(0, 40, 1000)
alpha = 0.025; omega = 0.75
x = complex_exponential_sequence(n, alpha, omega)

Plot of real and imaginary parts of the sequence generated for various values of \(\alpha\) and \(\omega\) is given next

Complex exponential sequence for various values of decay factor and angular frequency
Figure 1: Complex exponential sequence for various values of \(\alpha\) and \(\omega\)

From Figure 1, we see that the variable \(\alpha\) governs the decay or growth of the sequence in time and the \(\omega\) controls the oscillation frequency on a circle in the complex plane.

The 3D views of the complex sequence for various values of \(\alpha\) and \(\omega\) are illustrated next.

When (\(\alpha=0\)), the sequence remains the on circle in the complex plane.

A neutral complex exponential sequence

Figure 2: A neutral sequence (\(\alpha =0\) and \(\omega = 1\))

When \(\alpha \gt 0 \), the sequence grows exponentially and it spirals out.

A growing complex exponential sequence
Figure 3: A growing sequence (\(\alpha >0\) and \(\omega = 0.75 \))

When \(\alpha \lt 0 \), the sequence decays exponentially.

A decaying sequence
Figure 4: A decaying sequence (\(\alpha <0\) and \(\omega = 2.5 \))

Applications

Complex exponential sequences have various applications in modeling and signal processing. Some of the key applications include:

Signal Analysis and Representation: Complex exponential sequences form the basis for Fourier analysis, which decomposes a signal into a sum of sinusoidal components. The complex exponential sequence (\(e^{j\omega n}\)) serves as the building block for representing and analyzing signals in the frequency domain.

System Modeling and Analysis: These sequences play a fundamental role in modeling and analyzing linear time-invariant (LTI) systems. By applying complex exponential inputs to a system and observing the resulting outputs, one can determine the system’s frequency response and characterize its behavior in terms of amplitude and phase shifts at different frequencies.

Digital Filtering: Complex exponential sequences are utilized in digital filtering algorithms, such as the Fourier transform-based frequency domain filtering and the Z-transform-based discrete-time filtering. These sequences help design filters for various applications, such as noise removal, equalization, and signal enhancement.

Modulation Techniques: These sequences are fundamental in various modulation schemes used in communication systems. For instance, in amplitude modulation (AM), frequency modulation (FM), and phase modulation (PM), the modulating signals are typically expressed as complex exponential sequences that are mixed with carrier signals to encode information.

Control Systems: Complex exponential sequences are relevant in control system analysis and design. In control theory, the Laplace transform, which involves complex exponentials, is used to analyze system dynamics, stability, and transient response. The concept of the complex plane, where complex exponentials reside, is crucial in control system design and stability analysis.

Digital Signal Processing (DSP): These sequences find extensive use in various DSP applications, including digital audio processing, image processing, speech recognition, and data compression. Techniques like the discrete Fourier transform (DFT) and fast Fourier transform (FFT) exploit complex exponentials to efficiently analyze signals in the frequency domain.

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