Category Archives: Discrete-Time Signal Processing

Sinusoidal and Exponential Sequences, Periodicity of Sequences

Continuing our discussion on discrete-time sequences, we now come to define exponential and sinusoidal sequences. The general formula for a discrete-time exponential sequence is as follows:

x[n] = Aα^n.

This exponential behaves differently according to the value of α. If the sequence starts at n=0, the formula is as follows:

x[n] = Aα^n * u[n].

expo

If α is a complex number, the exponential function exhibits newer characteristics. The envelope of the exponential is |α|. If |α| < 1, the system is decaying. If |α|> 1, the system is growing.

cexpo

When α is complex, the sequence may be analyzed as follows, using the definition of Euler’s formula to express a complex relationship as a magnitude and phase difference.

Captu56 ma

Where ω0 is the frequency and φ is the phase, for n number of samples, a complex exponential sequence of form Ae^jw0n may be considered as a sinusoidal sequence for a set of frequencies in an interval of 2π.

A sinusoidal sequence is defined as follows:

x[n] = Acos(ω0*n + φ), for all n, and A, φ are real constants.

Periodicity for discrete-time signals means that the sequence will repeat itself for a certain delay, N.

x[n] = x[n+N] : system is periodic.

t = (-5:1:15)’;

impulse = t==0;
unitstep = t>=0;
Alpha1 = -0.5;
Alpha2 = 0.5;
Alpha3 = 2.5;
Alpha4 = -2.5;
cAlpha1 = -0.5 – 0.5i;
cAlpha2 = 0.5 + 0.5i;
cAlpha3 = 2.5 -2.5i;
cAlpha4 = -2.5 + 2.5i;
A = 1;

Exp1 = A.*unitstep.*Alpha1.^t;
Exp2 = A.*unitstep.*Alpha2.^t;
Exp3 = A.*unitstep.*Alpha3.^t;
Exp4 = A.*unitstep.*Alpha4.^t;

cExp1 = A.*unitstep.*cAlpha1.^t;
cExp2 = A.*unitstep.*cAlpha2.^t;
cExp3 = A.*unitstep.*cAlpha3.^t;
cExp4 = A.*unitstep.*cAlpha4.^t;

%%
figure(1)
subplot(2,1,1)
stem(t, impulse)
xlabel(‘x’)
ylabel(‘y’)
title(‘Impulse’)

subplot(2,1,2)
stem(t, unitstep)
xlabel(‘x’)
ylabel(‘y’)
title(‘Unit Step’)
%%
figure(2)
subplot(2,2,1)
stem(t, cExp1)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = -0.5 – 0.5i’)

subplot(2,2,2)
stem(t, cExp2)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = 0.5 + 0.5i’)

subplot(2,2,3)
stem(t, cExp3)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = 2.5 -2.5i’)

subplot(2,2,4)
stem(t, cExp4)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = -2.5 + 2.5i’)
%%
figure(3)
subplot(2,2,1)
stem(t, Exp1)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = -0.5’)

subplot(2,2,2)
stem(t, Exp2)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = 0.5’)

subplot(2,2,3)
stem(t, Exp3)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = 2.5’)

subplot(2,2,4)
stem(t, Exp4)
xlabel(‘n’)
ylabel(‘x[n]’)
title(‘Exponential: alpha = -2.5’)

 

 

Discrete-Time Impulse and Unit Step Functions

Discrete-Time Signals are understood as a set or sequence of numbers. These sequences possess magnitudes or values at a given index.

One mark of Discrete-Time Signals is that the index value is an integer. Thus, the sequence will have a magnitude or value for a whole number index such as -5, -4, 0, 6, 10000, etc.

A discrete-time signal represented as a sequence of numbers takes the following form:

x[n] = {x[n]},          -∞ < n < ∞,

where n is any real integer (the index).

An analog representation describes values of a signal at time nT, where T is the sampling period. The sampling frequency is the inverse of the sampling period.

x[n] = X_a(nT),      -∞ < n < ∞.

 

Common Sequences

Both a very simple and important sequence is the unit sample sequence, “discrete time impulse” or simply “impulse,” equal to 1 only at zero and equal to zero otherwise.

12

The discrete time impulse is used to describe an entire system using a delayed impulse. An entire sequence may also be shifted or delayed using the following relation:

y[n] = x[n – n0],

where n0 is an integer (which is the increment of indices by which the system is delayed. The impulse function delayed to any index and multiplied by the value of the system at that index can describe any discrete-time system. The general formula for this relationship is,

122

The unit step sequence is related to the unit impulse. The unit step sequence is a set of numbers that is equal to zero for all numbers less than zero and equal to one for numbers equal and greater than zero.

1222

The unit step sequence is therefore equal to a sequence of delta impulses with a zero and greater delay.

u[n] = δ[n] + δ[n-1] + δ[n-2] + . . .

12222

The unit impulse can also be represented by unit step functions:

δ[n] = u[n] – u[n-1].

Below I’ve plotted both the impulse and unit step function in matlab.

122222

t = (-10:1:10)';

impulse = t==0;
unitstep = t>=0;

figure(1)
subplot(2,1,1)
stem(t, impulse)
xlabel('x')
ylabel('y')
title('Impulse')
figure(1)
subplot(2,1,2)
stem(t, unitstep)
xlabel('x')
ylabel('y')
title('Unit Step')