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:
πΉ[n]: 1 at n=0, 0 otherwise.
Unit Step:
u[n] = 1 at n>=0, 0 otherwise.
Or,


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

Exponential & Sinusoidal Sequences
Exponential sequence: x[n] = A πͺn
where πͺ is complex, x[n] = |A|ejπ |πͺ|ejΟ0n=|A||πͺ|n ej(Ο0n+π)
= |A||πͺ|n(cos(Ο0n+π)+j sin(Ο0n+π))
Complex and sinusoidal: -π
< Ο0< π
or 0< Ο0< 2π
.
Exponential sequences for given πͺ (complex πͺ left, real πͺ right):

Periodicity: x[n] = x[n+N], for all n. (definition). Period = N.
Sinusoid: x[n] = A cos(Ο0n+π) = A cos (Ο0n+ Ο0N+ π)
Test: Ο0N = 2π
k, (k is integer)
Exponential: x[n] = ejΟ0(n+N) = ejΟ0n,
Test: Ο0N = 2π
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:
βResponseβ hk[n] describes how system behaves to impulse πΉ[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 convolutionβ¦
LTI is commutative: x[n]*h[n] = h[n]*x[n].
β¦ is additive: x[n]*(h1[n]+h2[n]) = x[n]*h1[n] + x[n]*h2[n].
β¦ is associative: (x[n]*h1[n])*h2[n] = x[n]*(h1[n]*h2[n])
LTI is stable if the sum of impulse responses

β¦ 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 = (πΉ[n+1]- πΉ[n])* πΉ[n-1] = πΉ[n] – πΉ[n-1].
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Continued: