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

Discrete-Time | Digital |

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**:

๐น[n]: 1 at n=0, 0 otherwise.

**Unit Step**:

u[n] = 1 at n>=0, 0 otherwise.

Or,

Any sequence: x[n] = a1* ๐น[n-1] + a2* ๐น[n-2]+โฆ

where a1, a2 are magnitude at integer n.

or,

### Exponential & Sinusoidal Sequences

Exponential sequence: x[n] = A ๐ช^{n}

where ๐ช is complex, x[n] = |A|e^{j๐} |๐ช|e^{jฯ}_{0}^{n}=|A||๐ช|^{n }e^{j(ฯ}_{0}^{n+๐)}

= |A||๐ช|^{n}(cos(ฯ_{0}n+๐)+*j *sin(ฯ_{0}n+๐))

Complex and sinusoidal: -๐
< ฯ_{0}< ๐
or 0< ฯ_{0}< 2๐
.

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

Periodicity: x[n] = x[n+N], for all n. (definition). Period = N.

Sinusoid: x[n] = A cos(ฯ_{0}n+๐) = A cos (ฯ_{0}n+ ฯ_{0}N+ ๐)

Test: ฯ_{0}N = 2๐
k, (k is integer)

Exponential: x[n] = e^{jฯ}_{0}^{(n+N) }= e^{jฯ}_{0}^{n},

Test: ฯ_{0}N = 2๐
k, (k is integer)

### System Properties

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

**Memoryless Systems:**

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

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

Additive property: Where y_{1}[n] = T{x_{1}[n]} and y_{2}[n] = T{x_{2}[n]},

y_{2}[n] + y_{1}[n] = T{x_{1}[n]+ x_{2}[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:

โResponseโ h_{k}[n] describes how system behaves to impulse ๐น[n-k] occurring at n = k.

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

Performing Discrete-Time convolution sum:

- Identify bounds of x[k] (where x[k] is non-zero) asN
_{1}and N_{2}. - Determine expression for x[k]h[n-k].
- Solve for

General solution for exponential (else use tables):

Graphical solution: superposition of responses h_{k}[n] for corresponding input x[n].

### LTI System Properties

As LTI systems are described by convolutionโฆ

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

โฆ is additive: x[n]*(h_{1}[n]+h_{2}[n]) = x[n]*h_{1}[n] + x[n]*h_{2}[n].

โฆ is associative: (x[n]*h_{1}[n])*h_{2}[n] = x[n]*(h_{1}[n]*h_{2}[n])

LTI is stable if the sum of impulse responses

โฆ 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: B_{h}=

If a<1, B_{h} is stable and (using geom. series) =

Delay on impulse response: h[n] = sequence*delay = (๐น[n+1]- ๐น[n])* ๐น[n-1] = ๐น[n] – ๐น[n-1].

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Continued: