Signals and Systems

Signals and Systems

Dr. Mohamed Bingabr

University of Central OklahomaSlides 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 operation• 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

• 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

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

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 signals2. Analogue and digital signals3. Periodic and aperiodic signals4. Energy and power signals5. Deterministic and probabilistic signals6. Causal and non-causal7. 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:

Signal Models – Unit Impulse Function

May use functions other than a rectangular pulse. Here are three example functions:

Exponential Triangular Gaussian

Note that the area under the pulse function must be unity

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.

The Exponential Function est

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


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 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)

Linear Systems (5)

Time-Invariant System

Instantaneous and Dynamic Systems

Causal and Noncausal Systems

Analogue and Digital Systems

Invertible and Noninvertible

Linear Differential Systems (1)


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

Signals and Systems

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