# Discrete-Time Signals and System Properties

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

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:
๐น[n]: 1 at n=0, 0 otherwise.
Unit Step:
u[n] = 1 at n>=0, 0 otherwise.

Or,

Any sequence: x[n] = a1* ๐น[n-1] + a2* ๐น[n-2]+โฆ
where a1, a2 are magnitude at integer n.
or,

### Exponential & Sinusoidal Sequences

Exponential sequence: x[n] = A ๐ชn
where ๐ช is complex, x[n] = |A|ej๐ |๐ช|ejฯ0n=|A||๐ช|n ej(ฯ0n+๐)
= |A||๐ช|n(cos(ฯ0n+๐)+j sin(ฯ0n+๐))
Complex and sinusoidal: -๐< ฯ0< ๐ or 0< ฯ0< 2๐.

Exponential sequences for given ๐ช (complex ๐ช left, real ๐ช right):

Periodicity:        x[n] = x[n+N],  for all n. (definition). Period = N.
Sinusoid: x[n] = A cos(ฯ0n+๐) = A cos (ฯ0n+ ฯ0N+ ๐)
Test: ฯ0N = 2๐k,                            (k is integer)

Exponential: x[n] = ejฯ0(n+N) = ejฯ0n,
Test: ฯ0N = 2๐k,                            (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).

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:

โResponseโ hk[n] describes how system behaves to impulse ๐น[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 convolutionโฆ

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

โฆ is additive: x[n]*(h1[n]+h2[n]) = x[n]*h1[n] + x[n]*h2[n].

โฆ is associative: (x[n]*h1[n])*h2[n] = x[n]*(h1[n]*h2[n])

LTI is stable if the sum of impulse responses

โฆ 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 = (๐น[n+1]- ๐น[n])* ๐น[n-1] = ๐น[n] – ๐น[n-1].

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Continued: