Discrete-Time Signals and System Properties

First, a comparison between Discrete-Time and Digital signals:

Discrete-TimeDigital
The independent variable (most commonly time) is represented by a sequence of numbers of a fixed interval. Both the independent variable and dependent variable are represented by a sequence of numbers of a fixed interval. 

Discrete-Time and Digital signal examples are shown below:

Discrete-Time Systems and Digital Systems are defined by their inputs and outputs being both either Discrete-Time Signals or Digital Signals.

Discrete-Time Signals

Discrete-Time Signal x[x] is sequence for all integers n.

  Unit Sample Sequence:
𝜹[n]: 1 at n=0, 0 otherwise.  
   Unit Step:
u[n] = 1 at n>=0, 0 otherwise.

Or,

Any sequence: x[n] = a1* 𝜹[n-1] + a2* 𝜹[n-2]+…
where a1, a2 are magnitude at integer n.
                                                or,           

Exponential & Sinusoidal Sequences

Exponential sequence: x[n] = A 𝞪n
                                                 where 𝞪 is complex, x[n] = |A|ej𝜙 |𝞪|e0n=|A||𝞪|n ej(ω0n+𝜙)
                                                                                                     = |A||𝞪|n(cos(ω0n+𝜙)+j sin(ω0n+𝜙))
                Complex and sinusoidal: -𝝅< ω0< 𝝅 or 0< ω0< 2𝝅.

                                                Exponential sequences for given 𝞪 (complex 𝞪 left, real 𝞪 right):

Periodicity:        x[n] = x[n+N],  for all n. (definition). Period = N.
                                Sinusoid: x[n] = A cos(ω0n+𝜙) = A cos (ω0n+ ω0N+ 𝜙)
                                                Test: ω0N = 2𝝅k,                            (k is integer)

                                Exponential: x[n] = e0(n+N) = e0n,
                                                Test: ω0N = 2𝝅k,                            (k is integer)

System Properties

                                  System: Applied transformation y[n] = T{x[n]}

Memoryless Systems:

                                Output y[nx] is only dependent on input x[nx] where the same index nx is used for both (no time delay or advance).

Linear Systems:               Adherence to superposition. The additive property and scaling property.

Additive property:         Where y1[n] = T{x1[n]} and y2[n] = T{x2[n]},
y2[n] + y1[n] = T{x1[n]+ x2[n]}.

Scaling property:            T{a.x[n]} = a.y[n]           

Time-Invariant Systems:

                                Time shift of input causes equal time shift of output. T{x[n-M]} = y[n-M]

Causality:

                                The system is causal if output y[n] is only dependent on x[n+M] where M<=0.

Stability:

                                Input x[n] and Output y[n] of system reach a maximum of a number less than infinity. Must hold for all values of n.

Linear Time-Invariant Systems

                                Two Properties: Linear & Time-Invariant follows:

                “Response” hk[n] describes how system behaves to impulse 𝜹[n-k] occurring at n = k.

  • Convolution Sum: y[n] = x[n]*h[n].

Performing Discrete-Time convolution sum:

  1. Identify bounds of x[k] (where x[k] is non-zero) asN1 and N2.
  2. Determine expression for x[k]h[n-k].
  3. Solve for

General solution for exponential (else use tables):

Graphical solution: superposition of responses hk[n] for corresponding input x[n].

LTI System Properties

As LTI systems are described by convolution…

LTI is commutative: x[n]*h[n] = h[n]*x[n].

                                … is additive: x[n]*(h1[n]+h2[n]) = x[n]*h1[n] + x[n]*h2[n].

                                … is associative: (x[n]*h1[n])*h2[n] = x[n]*(h1[n]*h2[n])

                LTI is stable if the sum of impulse responses

                                … is causal if h[n] = 0 for n<0                  (causality definition).

Finite-duration Impulse response (FIR) systems:

                Impulse response h[n] has limited non-zero samples. Simple to determine stability (above).

Infinite-duration impulse response (IIR) systems:

                Example: Bh=


If a<1, Bh is stable and (using geom. series) =

Delay on impulse response: h[n] = sequence*delay = (𝜹[n+1]- 𝜹[n])* 𝜹[n-1] = 𝜹[n] – 𝜹[n-1].

______________________________________________________

Continued:

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')