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:
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|e^j𝜙 |𝞪|e^jω₀ⁿ=|A||𝞪|ⁿ e^j(ω₀^n+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] = e^jω₀^(n+N) = e^jω₀ⁿ,
                                                Test: ω₀N = 2𝝅k,                            (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 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” h_k[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 h_k[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: B_h=

If a<1, B_h 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].

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Continued: