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 ๐ชn
where ๐ช is complex, x[n] = |A|ej๐ |๐ช|ejฯ0n=|A||๐ช|n ej(ฯ0n+๐)
= |A||๐ช|n(cos(ฯ0n+๐)+j sin(ฯ0n+๐))
Complex and sinusoidal: -๐
< ฯ0< ๐
or 0< ฯ0< 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(ฯ0n+๐) = A cos (ฯ0n+ ฯ0N+ ๐)
Test: ฯ0N = 2๐
k, (k is integer)
Exponential: x[n] = ejฯ0(n+N) = ejฯ0n,
Test: ฯ0N = 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).
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:
โResponseโ hk[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) asN1 and N2.
- Determine expression for x[k]h[n-k].
- 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 convolutionโฆ
LTI is commutative: x[n]*h[n] = h[n]*x[n].
โฆ is additive: x[n]*(h1[n]+h2[n]) = x[n]*h1[n] + x[n]*h2[n].
โฆ is associative: (x[n]*h1[n])*h2[n] = x[n]*(h1[n]*h2[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: Bh=

If a<1, Bh 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].
______________________________________________________
Continued: