The Geometric Series in Digital Signal Processing

In Matlab an array operator “.ˆ” is required to implement a real exponential sequence of the form:

Over the n₁< n_o < n₂ interval. For example, to implement:

We will use the following Matlab function:

n=[0:10];;
x=(0.9).^n;
stem(n,x);
xlabel(‘n’); ylabel(‘x[n]’)

This script yields:

Geometric series

A one-side exponential sequence of the form:

Where α is an arbitrary constant. This sequences is called a geometric series.  In DSP, the convergence and expression for the sum of this series are used in many applications. The series converges for:

While this condition is true, the sum of the geometric series components converges to:

From here, we also need an expression for the sum of any finite number of terms of the series, and that is given by:

These two important results will be used deeply throughout DSP.

Complex-valued exponential sequence

Where σ produces an attenuation (if<0) or amplification (if>0)  and ω_o is the frequency in radians. The Matlab function exp generates the exponential sequences. For example, for x_[n] =exp[(2+j3)n], 0≤n≤10, we will use the following script:

n=[0:10]; x=exp(3j*n);
stem(n,k)

This script yields:

Source:

• Digital Signal Processing Using Matlab, 3erd ed

PREVIOUS:

• What is DSP ?- Digital Signal Processing
•  Matlab code for DSP – Introduction
• Elementary sequences in Digital Signal Processing
• The sinusoidal function in discrete-time (Versión en Español)
• Convolution in Discrete-Time in Matlab (Versión en Español)
• Solving discrete-time differential equations with Matlab (Versión en Español)
• Digital Filters – Moving Average Filters (Versión en Español)
• The Discrete-Time Fourier Transform (Versión en Español)

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