Signals and Systems (C010-6)

Assist. Prof. Dr. Rong Ran

sunnyran@ajou.ac.kr

2015.fall

What is a signal?

A signal is a quantitative description of a physical phenomenon, event or process. Electrical current or voltage in a circuit .

Daily closing value of share of stock last week.

Audio signal.

A signal is a function, usually of one variable of time. However, in general, signal can be functions of more than one variable, e.g., image signals.

Signal Types

In this class we are interested in two types of signals: Continuous-time signal x(t), where t is a real-valued variable denoting time.

We use parenthesis (.) to denote a continuous-time signal.

Discrete-time signal x[n], where n is an integer-valued variable denoting the discrete samples of time. We use square brackets [.] to denote a discrete-time signal.

Review on complex numbers

We are interested in the general complex signals:

Cont.

Basic operation of signals

Time shift

Time reversal

Cont.

Time Scaling

Combination of operations

Cont. Several examples:

Decimation & Expansion

Two standard discrete-time signal processing operations Decimation

M is called the decimation factor.

Expansion

L is called the expansion factor

Periodicity

Definition

Question 1

Periodic or Aperiodic ?

Question 2

Suppose x(t) is periodic. Is y(t)=x(at) periodic if a>0? And what is the fundamental period of y(t)?

Suppose x[n] is periodic. Is y[n]=x[mn] periodic ? And what is the fundamental period of y[n]?

Even and odd signals

Definition A continuous-time signal x(t) is even if x(-t)=x(t), and it is odd if x(-t)=-x(t);

A discrete-time signal x[n] is odd if x[-n]=x[n], and it is odd if x[-n]=-x[n].

Discrete-time Impulse and step func-tions

The discrete-time unit impulse signal is defined as

The discrete-time unit step signal is defined as

It can be shown that

Property of Impulse function

Sampling property

Shifting property

Representation property

Continuous-time impulse and step function

The dirac delta function is defined as

The unit step function is defined as

Property of Dirac delta function

Sampling property

Shifting property

Representation property

Fundamentals of systems

A system is a quantitative description of a physical process which transforms signals (“input”） to signals (“output”).

System Properties

Memoryless : A system is memoryless if the output at time t (or n) depends only on the input at time t (or n). y(t)=2x(t) ?

y[ n]=4x[n]?

y[n]=x[n-1]?

Invertible : A system is invertible if distinct input signals produce distinct output signals To show that a system is invertible, one has to show the inversion formula

To show that a system is not invertible, one has to given a counter example

Cont. Example 1

Example 2

Cont.

Causal : A system is causal if the output at time t (or n) depends only on inputs at time (s<=t) (i.e., the present and past) y[n]=x[n-1] ?

y[n]=x[n]+x[n-2] ?

y[n]=x[-n] ?

y(t)=x(t)cos(t+1) ? Memoryless?

Stable

Cont.

Example 1

Cont.

Time-invariant : A system is time-invariant if a time-shift of the input signal results in the same time-shift of the output signal. That is, if

Cont.

Example 1

Cont.

Linear: A system is linear if it is additive and scalable, that is

ax1(t)+bx2(t)ay1(t)+by2(t) for all a and b. Example 1:

Example 2: