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Elementary sequences in digital signal processing with Matlab

In MATLAB we can represent a finite-duration sequence by a row vector of appropriate values. However, such a vector does not have any information about sample position n. therefore, a correct representation of a discrete function x[n] would require two vectors, one each for x and n. For example, a sequence x[n] ={2,1,-1,0,1,4,3,7} can be represented in Matlab by:

>> n=[-3,-2,-1,0,1,2,3,4]; x=[2,1,-1,0,1,4,3,7]

Generally, we will use the x-vector representation alone when the sample position information is not required or when such information is trivial (e.g. when the sequence begins at n=0)

Types of sequences

We use several elementary sequences in Digital Signal Processing for analysis purposes.

Unit Sample sequence

The function zeros(1,N) generates a row vector of N zeros, which can be used to implement δ[n] over a finite interval. However, the logical relation n==0 is an elegant way of implementing δ[n]. For example, to implement:

Over the n1< no < n2  interval, will use the following Matlab function:

function[x,n]=impseq(n0,n1,n2)

%generates x[n]=delta(n-no); n1<=n<=n2

n=[n1:n2]; x=[(n-n0)==0];

Now, we can use the impseq function as follows, to implement an arbitrary function T=δ[n-5] over the 0< no < 9  interval:

>> T=impseq(5,0,9)

Result:

T =     0     0     0     0     0     1     0     0     0     0

Note: to see how to implement a Matlab function see: matlab getstart, page 176, (4-22)

The power of this approach can be see in the follow example.

Example 1. Generate and plot the following sequence over the indicated interval:

n=[-5:5];
x=2*impseq(-2,-5,5)-impseq(4,-5,5);
stem(n,x); title(‘Secuencia de Problema 1’)
xlabel(‘n’); ylabel(‘x[n]’)

This commands yield:

Unit Step sequence

The function ones(1,N) generates a row vector of N ones. It can be used to generate u[n] over a finite interval. Once again, an elegant approach is to use the logical relation n>=0. To implement:

Over the n1< no < n2  interval, will use the following Matlab function:

function[x,n]=stepseq(n0,n1,n2)

% Generates x(n)=u(n-n0); n1<=n<=n2

n=[n1:n2]; x=[(n-n0)>=0];

Now, we can use the stepseq function as follows, to implement an arbitrary function P=u[n-5] over the 0< no < 18 interval:

>> P=stepseq(5,0,18)

Result:

P =     0     0     0     0     0     1     1     1     1     1     1     1     1     1     1     1     1     1     1

Example 2. Generate and plot each of the following sequence over the indicated interval:

n=[0:20]; x1=10*exp(-0.3*(n-10)).*(stepseq(10,0,20)-stepseq(20,0,20));

x2=n.*(stepseq(0,0,20)-stepseq(10,0,20));
x=x1+x2;
stem(n,x); title(‘Secuencia de Problema 2’)

This commands yield:

Real-valued exponential sequence

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

Over the n1< no < 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 ωo is the frequency in radians. The Matlab function exp generates the exponential sequences. For example, for x[n] =exp[(2+j3)n], 0n10, we will use the following script:

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

This script yields:

Sinusoidal sequence 

Sine waves are important because Fourier´s Theorem states that most signals of practical interest can be decomposed into an infinite sum of sine waves. Discrete-time signals (also called time series) are defined over the set of integers, that is, they are indexed sequences. A discrete-time sine wave is defined by:

Where A is an amplitude and  θo is the phase in radians. Where A is an amplitude and teta is the phase in radians. Meanwhile, ωo=2πf is the angular frequency and x[n] could be written as:

It is important to understand that the frequency of a discrete-time sinusoid is not uniquely defined. This fundamental ambiguity is a consequence of a basic trigonometric property:

In words, the value of a sinusoid does not change if an integer multiple of is added to its argument. Adding the 2πkn to the argument of equation (1) we get:

Two cases must be distinguished. If k≥-f, the equation (2) is equivalent to a sinusoid with frequency f+k with no change in phase:

On the other hand, if k<-f, equation (3) leads to a negative frequency. To avoid this, we introduce:

We also make use of the property:

In consequence, returning to equations (2) and three, we obtain a sinusoid of frequency l-f with a reversal in phase:

In conclusion, a discrete-time sinusoid with frequency f is identical to a same-phase sinusoid of frequency f+k, where k is any integer greater than –f, or to a phase-reversed sinusoid of frequency l-f if l>f.

Equation (3) can be expressed more concisely using complex exponential notation:

Because value of a complex exponential does not change if a multiple of is added to its argument, we get:

Equation (5) is equivalent to equation (4). Because of this fundamental frequency ambiguity, we will often implicitly assume that the angular frequency of a discrete-time sinusoid is restricted to the range –π≤ω≤π, or equivalent, that -1/2≤f≤1/2.

The Matlab function cos or sin generates the sinusoidal sequences. For example, for x[n]=3cos(0.1πn+π/3)+2sin(0.5πn), 0n10, we will use the following script:

n=[0:10]; x=3*cos(0.1*pi*n+pi/3)+2*sin(0.5*pi*n);
stem(n,x)

This script yields:

In construction ….

Source:

  • Digital Signal Processing Using Matlab, 3erd ed
  • Fundamentos_de_Señales_y_Sistemas_usando la Web y Matlab
  • Oppenheim – Señales y Sistemas
  • Análisis de Sistemas Lineales Asistido con Scilab – Un Enfoque desde la Ingeniería Eléctrica.

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