Signals and Systems

Dr. Mohamed Bingabr

University of Central OklahomaSome of the Slides For Lathi’s Textbook Provided by

Dr. Peter Cheung

Course Objectives

• Signal analysis (continuous-time)• System analysis (mostly continuous systems)• Time-domain analysis (including convolution)• Laplace Transform and transfer functions• Fourier Series analysis of periodic signal• Fourier Transform analysis of aperiodic signal• Sampling Theorem and signal reconstructions

Outline

• Size of a signal• Useful signal operations• Classification of Signals• Signal Models• Systems• Classification of Systems• System Model: Input-Output Description• Internal and External Description of a System

Size of Signal-Energy Signal

• Signal: is a set of data or information collected over time.

• Measured by signal energy Ex:

• Generalize for a complex valued signal to:

• Energy must be finite, which means

Size of Signal-Power Signal

• If amplitude of x(t) does not 0 when t ", need to measure power Px instead:

• Again, generalize for a complex valued signal to:

Useful Signal Operation-Time Delay

Find x(t-2) and x(t+2) for the signal

elsewhere

ttx

0

412)(

1 4

2

t

x(t)

Useful Signal Operation-Time Delay

Signal may be delayed by time T:

(t) = x (t – T)

or advanced by time T:

(t) = x (t + T)

Useful Signal Operation-Time Scaling

Find x(2t) and x(t/2) for the signal

elsewhere

ttx

0

412)(

1 4

2

t

x(t)

Useful Signal Operation-Time Scaling

Signal may be compressed in time (by a factor of 2):

(t) = x (2t)

or expanded in time (by a factor of 2):

(t) = x (t/2)

Same as recording played back attwice and half the speedrespectively

Useful Signal Operation-Time Reversal

Signal may be reflected about the vertical axis (i.e. time reversed):

(t) = x (-t)

Useful Signal Operation-Example

We can combine these three operations.

For example, the signal x(2t - 6) can be obtained in two ways;

• Delay x(t) by 6 to obtain x(t - 6), and then time-compress thissignal by factor 2 (replace t with 2t) to obtain x(2t - 6).

• Alternately, time-compress x(t) by factor 2 to obtain x(2t), then delay this signal by 3 (replace t with t - 3) to obtain x(2t - 6).

Signal Classification

Signals may be classified into:

1. Continuous-time and discrete-time signals 2. Analogue and digital signals 3. Periodic and aperiodic signals 4. Energy and power signals 5. Deterministic and probabilistic signals 6. Causal and non-causal 7. Even and Odd signals

Signal Classification- Continuous vs Discrete

Continuous-time

Discrete-time

Signal Classification- Analogue vs Digital

Analogue, continuous

Analogue, discrete

Digital, continuous

Digital, discrete

Signal Classification- Periodic vs Aperiodic

A signal x(t) is said to be periodic if for some positive constant To

x(t) = x (t+To) for all t

The smallest value of To that satisfies the periodicity condition of this equation is the fundamental period of x(t).

Signal Classification- Deterministic vs Random

Deterministic

Random

Signal Classification- Causal vs Non-causal

Signal Classification- Even vs Odd

Signal Models – Unit Step Function u(t)

Step function defined by:

Useful to describe a signal that begins at t = 0 (i.e. causal signal).

For example, the signal e-at represents an everlasting exponential that starts at t = -.

The causal for of this exponential e-atu(t)

Signal Models – Pulse Signal

A pulse signal can be presented by two step functions:

x(t) = u(t-2) – u(t-4)

Signal Models – Unit Impulse Function δ(t)

First defined by Dirac as:

Multiplying Function (t) by an Impulse

Since impulse is non-zero only at t = 0, and (t) at t = 0 is (0), we get:

We can generalize this for t = T:

Sampling Property of Unit Impulse Function

Since we have:

It follows that:

This is the same as “sampling” (t) at t = 0.If we want to sample (t) at t = T, we just multiple (t) with

This is called the “sampling or sifting property” of the impulse.

Examples

Simplify the following expression

)3(2

1

j

Evaluate the following

dtet t)3(

Find dx/dt for the following signal

x(t) = u(t-2) – 3u(t-4)

The Exponential Function est

This exponential function is very important in signals & systems, and the parameter s is a complex variable given by:

The Exponential Function est

If = 0, then we have the function ejωt, which has a real frequency of ω

Therefore the complex variable s = +jω is the complex frequency

The function est can be used to describe a very large class of signals and functions. Here are a number of example:

The Exponential Function est

The Complex Frequency Plane s= + jω

A real function xe(t) is said to be an even function of t if

A real function xo(t) is said to be an odd function of t if

HW1_Ch1: 1.1-3, 1.1-4, 1.2-2(a,b,d), 1.2-5, 1.4-3, 1.4-4, 1.4-5, 1.4-10 (b, f)

Even and Odd Function

Even and odd functions have the following properties:• Even x Odd = Odd• Odd x Odd = Even• Even x Even = Even

Every signal x(t) can be expressed as a sum of even andodd components because:

Even and Odd Function

Consider the causal exponential function

What are Systems?

• Systems are used to process signals to modify or extract information

• Physical system – characterized by their input-output relationships

• E.g. electrical systems are characterized by voltage-current relationships

• From this, we derive a mathematical model of the system• “Black box” model of a system:

Classification of Systems

Systems may be classified into:

1. Linear and non-linear systems2. Constant parameter and time-varying-parameter systems3. Instantaneous (memoryless) and dynamic (with memory)

systems4. Causal and non-causal systems5. Continuous-time and discrete-time systems6. Analogue and digital systems7. Invertible and noninvertible systems8. Stable and unstable systems

Linear Systems (1)

• A linear system exhibits the additivity property: if and then

• It also must satisfy the homogeneity or scaling property: if then

• These can be combined into the property of superposition: if and then

• A non-linear system is one that is NOT linear (i.e. does not obey the principle of superposition)

Linear Systems (2)

Linear Systems (3)

Linear Systems (4)

Is the system y = x2 linear?

Linear Systems (5)

A complex input can be represented as a sum of simpler inputs (pulse, step, sinusoidal), and then use linearity to find the response to this simple inputs to find the system output to the complex input.

Time-Invariant System

Which of the system is time-invariant? (a) y(t) = 3x(t) (b) y(t) = t x(t)

Instantaneous and Dynamic Systems

Causal and Noncausal Systems

Which of the two systems is causal? a) y(t) = 3 x(t) + x(t-2)b) y(t) = 3x(t) + x(t+2)

Analogue and Digital Systems

Invertible and Noninvertible

Which of the two systems is invertible?a) y(t) = x2

b) y= 2x

System External Stability

Electrical System

)()( tiRtv dt

dvCti )(

dt

diLtv )(

Ri(t)

+ v(t) -

i(t) + v(t) -

+ v(t) -i(t)

Mechanical System

2

2

)()(dt

ydMtyMtx )()( tyktx

dt

dyBtyBtx )()(

Linear Differential Systems (1)

Linear Differential Systems (2)

Find the input-output relationship for the transational mechanical system shown below. The input is the force x(t), and the output is the mass position y(t)

Linear Differential Systems (3)

Linear Differential Systems (4)

HW2_Ch1: 1.7-1 (a, b, d), 1.7-2 (a, b, c), 1.7-7, 1.7-13, 1.8-1, 1.8-3