Signals and Systems - System Classifications

What is a System?

A system processes an input signal $x(t)$ to produce an output signal $y(t)$ .
Broadly, a system is anything that responds when stimulated.
A basic system can be represented as a block diagram with an input signal and an output signal, connected by a system function $h(t)$ .

Examples of Systems (Block Diagrams)

Integrator
Adder
Multiplier

A. Continuous-Time and Discrete-Time Systems

Continuous Time System

Operates with a continuous-time signal and produces a continuous-output signal.
A continuous-time signal $x(t)$ is transformed into another continuous-time signal $y(t)$ .

Discrete-Time System

Operates with a discrete-time signal and produces a discrete-output signal.
A discrete-time signal $x[n]$ is transformed into another discrete-time signal $y[n]$ .

B. Systems With Memory and Without Memory

Memoryless (Static) System

The output at any instant depends only on the input at that instant, not on past input values.

System with Memory (Dynamic)

The output at any instant depends on past or future input values.
All differential equations represent systems with memory.
Systems with memory are also called Dynamic systems.
Memoryless systems are also called Instantaneous, Static, or Memoryless systems.

Memory vs. Memoryless Comparison

Memory System: Present output depends on past or future input.
Memoryless System: Present output depends only on present input.

Procedure for Testing Memory

Choose a value for $t$ (or $n$ ).
Substitute this value into the input.
Determine if the input values are equal to, more than, or less than the present output value.
Justify whether the system has memory based on the characteristics (past, present, or future input).

Example

If the present output is assumed to be 5:
- Past input value: < 5
- Present input value: 5
- Future input value: > 5

EXAMPLE 1: Memory and Memoryless System

(a)

Moving-average system: $y[n] = (1/3)(x[n-1] + x[n] + x[n+1])$
It has memory because the output $y[n]$ at time $n$ depends on the present and two past values of $x[n]$ .

(b)

$y[n] = 2x[n]$
It is memoryless because the output signal $y[n]$ depends only on the present value of the input signal $x[n]$ .

EXAMPLE 2

System: $y(t) = 7x(t) + 6$
Let $t = 2$ : $y(2) = 7x(2) + 6$
Since the present output depends only on the present input value, the system is memoryless.

EXAMPLE 3

System: $y(t) = 7x(t+1) + 6$
Let $t = 2$ : $y(2) = 7x(3) + 6$
Since the present output value depends on a future input, the system has memory.

C. Causal and Noncausal Systems

Causal System

A system is causal if its output depends only on present and past values of the input, but not on future values.

Noncausal System

A system is non-causal if its output depends on future values of the input.

Example: Causal and Noncausal

$y[n] = x[n+1]$ is noncausal because the output signal $y[n]$ depends on a future value of the input signal, $x[n+1]$ .
Causality is required for a system to operate in real time.

Procedures for Testing Causality

The procedure is the same as with memory:
1. Choose a value for $t$ .
2. Substitute $t$ into the input.
3. Determine if the input values are equal to, more, or less than the present output value.
4. Justify whether the system is causal or non-causal based on the characteristics.

Causal vs. Non-causal Comparison

Causal System: Present output values depend on past or present input values.
Non-causal System: Present output values depend on future input.

EXAMPLE 1

System: $y(t) = 7x(t) + 6$
Let $t = 2$ : $y(2) = 7x(2) + 6$
Since present output depends on the present input value, the system is causal.

EXAMPLE 2

System: $y(t) = 8x(t) + 7x(t+5)$
Let $t = 2$ : $y(2) = 8x(2) + 7x(7)$
Since the present output depends on a future input value, the system is non-causal.

D. Linear Systems and Nonlinear Systems

Linear System Conditions

If an operator $T$ satisfies the following two conditions, then $T$ is a linear operator, and the system represented by $T$ is a linear system:
1. Additivity: Given that $T{x1} = y1$ and $T{x2} = y2$ , then $T{x1 + x2} = y1 + y2$
2. Homogeneity (or Scaling): If $T{x} = y$ , then $T{ax} = ay$

Superposition Principle

A system is linear if superposition (additivity and homogeneity) can be applied. Superposition = additivity + homogeneity.
Superposition implies that the response resulting from several input signals can be computed as the sum of the responses resulting from each input signal acting alone.
A system is non-linear if superposition cannot be applied.

Linear vs. Non-linear Comparison

Linear System:
- Superposition can be applied.
- Both Additivity AND Homogeneity must be true.
Non-linear System:
- Superposition cannot be applied.
- Either Additivity OR Homogeneity is not true.

Procedure for Testing Linearity

Let $y1(t)$ be the output corresponding to the input $x1(t)$ , and $y2(t)$ be the output corresponding to the input $x2(t)$ .
Additivity Property: The response to $x1(t) + x2(t)$ is $y1(t) + y2(t)$ .
Homogeneity Property: The response to $\alpha x1(t)$ is $\alpha y1(t)$ , where $\alpha$ is any arbitrary constant.

The two properties can be combined into a single statement: $\alpha x1(t) + \beta x2(t) \rightarrow \alpha y1(t) + \beta y2(t)$
- Where $\alpha$ and $\beta$ are arbitrary constants.

EXAMPLE 1

System: $y(t) = tx(t)$
Solution:
- Let $x1(t) \rightarrow y1(t) = tx_1(t)$
- Let $x2(t) \rightarrow y2(t) = tx_2(t)$
- Let $x3(t) = x1(t) + x2(t) \rightarrow y3(t) = tx_3(t)$
- Additivity Test:
 - $x3(t) = x1(t) + x_2(t)$
 - $y3(t) = t(x1(t) + x2(t)) = tx1(t) + tx2(t) = y1(t) + y_2(t)$
- Homogeneity Test:
 - Let $x3(t) = ax1(t)$
 - $y3(t) = t(ax1(t)) = a(tx1(t)) = ay1(t)$
- Therefore, the system is linear.

Linear Systems and Nonlinear Systems

Linear Systems: Systems that follow the principle of superposition.
1. Law of Additivity:
 - If $x1(t) \rightarrow y1(t)$ and $x2(t) \rightarrow y2(t)$ , then $x1(t) + x2(t) \rightarrow y1(t) + y2(t)$ .
2. Law of Homogeneity:
 - If $x(t) \rightarrow y(t)$ , then $kx(t) \rightarrow ky(t)$ .

E. Time-Invariant and Time-Variant Systems

Time-Invariant System

A system is time-invariant if a time shift (delay or advance) in the input signal causes the same time shift in the output signal.
Continuous-time: If $T{x(t - t0)} = y(t - t0)$ for any real value of $t_0$ .
Discrete-time: If $T{x[n - k]} = y[n - k]$ for any integer $k$ .

Time-Varying System

A system that does not satisfy the above equations is a time-varying system.

Time-Variant vs. Time-Invariant

Time-Variant: Input-output relationship varies with time; the same input produces different outputs at different times.
- $y2(t) \neq y1(t - t_0)$
Time-Invariant: The input-output relationship does not vary with time; the same input produces the same output at different times.
- $y2(t) = y1(t - t_0)$

Procedure for Testing Time-Invariance

Let $y1(t)$ be the output corresponding to the input $x1(t)$ .
Consider a second input, $x2(t)$ , obtained by shifting $x1(t)$ : $x2(t) = x1(t - t_0)$ .
Find the output $y2(t)$ corresponding to the input $x2(t)$ .
Find $y1(t - t0)$ from step 1 and compare with $y_2(t)$ .
If $y2(t) = y1(t - t_0)$ , then the system is time-invariant; otherwise, it is time-variant.

EXAMPLE 1

System: $y(t) = \sin(x(t))$
Let $x1(t) \rightarrow y1(t) = \sin(x_1(t))$
Let $x2(t) = x1(t - t_0)$
Then $y2(t) = \sin(x2(t)) = \sin(x1(t - t0))$
$y1(t - t0) = \sin(x1(t - t0))$
Since $y2(t) = y1(t - t_0)$ , the system is time-invariant.

EXAMPLE 2

System: $y(t) = tx(t)$

Let $x1(t) \rightarrow y1(t) = tx_1(t)$
Introduce a time delay $t0$ in the input: $x2(t) = x1(t-t0)$
The delayed input produces an output: $y2(t) = tx1(t-t_0)$ .
Introduce a time delay $t0$ in the output of equation (1): $y1(t-t0) = (t-t0)x1(t-t0)$
Compare equations (2) and (3): $y2(t) \neq y1(t - t_0)$ . Therefore, the system is Time-variant.

HOMEWORK

Check Time-Invariance and Time-Variance Properties of the following systems:
1. $y(t) = x(\sin(t))$
2. $y(t) = x(t + 2)$
3. $y(t) = \cos(t) + x(t)$

F. Linear Time-Invariant (LTI) Systems

If a system is both linear and time-invariant, it is called a linear time-invariant (LTI) system.
Linear: $\alpha x1(t) + \beta x2(t) \rightarrow \alpha y1(t) + \beta y2(t)$
Time-invariant: $y2(t) = y1(t - t_0)$
Why LTI? For an LTI system, knowing the impulse response $h(t)$ is sufficient to predict the output for any input by convolving the input with the impulse response. LTI systems are easier to analyze, which is not true for nonlinear or time-variant systems.

G. Stable Systems

A system is stable if a bounded-input produces a bounded-output (BIBO).
Bounded input refers to a finite value of the input signal $x(t)$ for any value of $t$ .
If $x(t)$ is bounded, then there exists a constant M < \infty such that |x(t)| \le M < \infty for all $t$ .
Stable: If its impulse response $h(t)$ vanishes after a sufficiently long time (goes to 0, convergence).
Unstable: If its impulse response $h(t)$ grows without bound (approaches $\infty$ ) after a sufficiently long time (divergence).

H. Feedback Systems

A feedback system is a special class of systems where the output signal is fed back and added to the input.
In a feedback system, the output at any time depends on past output, past input, and present input.
In a non-feedback system, the output depends only on the present and past input.

Interconnection of Systems

(a) Series (Cascade): $S1 \rightarrow S2$
(b) Parallel: $S1 + S2$
(c) Feedback: $S1, S2, S_3$