1Lecture No. 1

What is a Signal?• Signals = variables• A signal is a function of one or more

independent variables that conveys information on the nature of a physical phenomenon.

• A quantitative description of a variable changing in time (or space)

• Contains a pattern of variations

Introduction to Signals and Systems- Signals and Systems

2

Signal Examples• Internet: information-bearing signals• Listening to the heartbeat of a patient or

monitoring blood pressure and temperature. Both cases signals convey information to the doctor about the state of the patient.

• Human voice mechanism produces speech• Fluctuations in acoustic pressure• Acoustic pressure as a function of time

3

What is a System?

• System = operations• An operator acting among variables. • Set of interconnected operations that

perform a certain task.• A system is an entity that manipulates one

or more signals to accomplish a function, thereby yielding new signals.

4

Systems

• An automatic speaker recognition system: the input signal = a speech (voice), the system = computer, the output = the identity of the speaker.

• A blood pressure monitoring system: the input signal = sensor measurement, the system = computer, the output = blood pressure display.

5

A system

2x(t) 1/2y(t)x(t)y(t)=2x(t) z(t)=1/2[2x(t)]=x(t)

6

Communication Systems

(a) Snapshot of Pathfinderexploring the surface of Mars.

(b) The 70-meter (230-foot) diameter antenna located at Canberra, Australia. The surface of the 70-meter reflector must remain accurate within a fraction of the signal’s wavelength. (Jet Propulsion Laboratory.)

7

Control Systems

Block diagram of a feedback control system. The controller drives the plant, whose disturbed output drives the sensor(s). The resulting feedback signal is subtracted from the reference input to produce an error signal e(t), which, in turn, drives the controller. The feedback loop is thereby closed.

8

Control Systems

NASA space shuttle launch.(Courtesy of NASA.)

9

Biomedical Signals

Morphological types of nerve cells (neurons) identifiable in monkey cerebral cortex, based on studies of primary somatic sensory and motor cortices. (Reproduced from E. R. Kande, J. H. Schwartz, and T. M. Jessel, Principles of Neural Science, 3d ed., 1991; courtesy of Appleton and Lange.)

10

Biomedical Signals

The traces shown in (a), (b), and (c) are three examples of EEG signals recorded from the hippocampus of a rat. Neurobiological studies suggest that the hippocampus plays a key role in certain aspects of learning and memory.

11

Classification of Signals1. Continuous (or analog)-time vs. discrete-time signals

(a) Continuous-time signal x(t). (b) Representation of x(t) as a discrete-time signal x[n].

- Signal Classification

12

Classification of Signals2. Even vs. Odd Signals• Even Signal: x(-t) = x(t) for all t.• Odd Signal: x(-t) = -x(t) for all t.

13

Classification of Signals3. Periodic signals vs. Non-periodic signals• A periodic signal x(t)

– If x(t) = x(t+T) for all t,– T = fundamental period of x(t)– f=1/T=fundamental frequency

• If no value of T satisfies the condition above, a signal x(t) is called aperiodic or non-periodic.

Question: most biomedical signals are periodic or non-periodic?

14

(a) Square wave with amplitude A = 1 and period T = 0.2s. (b) Rectangular pulse of amplitude A and duration T1.

15Lecture No. 1

Classification of Signals4. Deterministic Signals vs. Random (or stochastic) Signals• A deterministic signal is a signal about which there is no

uncertainty with respect to its value at any time.• Deterministic signals are repeatable. That is when we

measure some values over a specified period of time and duplicate the measurement some time later, the same signal values occur.

• A random signal is a signal about which there is uncertainty before it occurs.

• If a signal does not achieve the same values, then it is no longer repeatable and considered random (or stochastic)

Question: how do we handle random or stochastic signals?

16Lecture No. 1

Classification of Signals

5. Energy of Signals and Power of Signals

Instantaneous power of the signal 2( ) ( )p t x tTotal Energy 2 ( )E x t dt

Average Power/ 2 2

/ 2

1 ( )T

TP x x dt

T

Root Mean-square (rms) Power/ 2 2

/ 2

1 ( )T

TP x x dt

T

How to size a signal? Consider amplitude and duration

t

t

17Lecture No. 1

Basic Operations on Signals• Operations performed on dependent variables1. Amplitude Scaling

( ) ( )[ ] [ ]

y t cx ty n cx n

where c is the scaling factor.

• A physical example of a device that performs amplitude scaling is an electronic amplifier.

- Basic Signal Operations

18Lecture No. 1

Basic Operations on Signals• Operations performed on dependent variables2. Addition

1 2

1 2

( ) ( ) ( )[ ] [ ] [ ]

y t x t x ty n x n x n

• A physical example: an audio mixer.

19Lecture No. 1

Basic Operations on Signals• Operations performed on dependent variables3. Multiplication

1 2

1 2

( ) ( ) ( )[ ] [ ] [ ]

y t x t x ty n x n x n

• A physical example: an AM radio signal where x1consists of an audio signal plus a dc component and x2 consists of a sinusoidal signal called a carrier wave.

20

AM vs. FM

21Lecture No. 1

Basic Operations on Signals• Operations performed on dependent variables4. Differentiation

( ) ( )dy t x tdt

Inductor with current i(t), inducing voltage v(t) across its terminals.

( ) ( )dv t L i tdt

22Lecture No. 1

Basic Operations on Signals• Operations performed on independent variables1. Time Scaling ( ) ( )

[ ] [ ], 0y t x aty n x kn k

If a>1, the signal y(t) is compressed.If 0<a<1, the signal y(t) is expanded.

23Lecture No. 1

Basic Operations on Signals• Operations performed on independent variables2. Reflection

( ) ( )y t x t

24Lecture No. 1

Basic Operations on Signals• Operations performed on independent variables3. Time Shifting

0( ) ( )y t x t t

25Lecture No. 1

Elementary Signals1. Exponential Signals

( ) atx t Be• B and a are real parameters

• B = amplitude at t=0

• if a < 0, decaying exponential

• if a > 0, growing exponential

• Exponential Signals: in discrete time

[ ] ,n ax n Br r e

- Elementary Signals

First given by Euler

26Lecture No. 1

Ex.: Exponential Signals

(a) Decaying exponential form of continuous-time signal. (b) Growing exponential form of continuous-time signal.

27Lecture No. 1

Ex.: Exponential Signals

(a) Decaying exponential form of discrete-time signal. (b)

Growing exponential form of discrete-time signal.

28Lecture No. 1

Elementary Signals2. Sinusoidal Signals

( ) cos( )x t A t • A = amplitude

• = frequency in radians per second

• = phase angle in radians

• Sinusoidal signals are periodic

x(t)=x(t + T)

Jyā, koti-jyā and utkrama-jyā are three trigonometric functions introduced by Indian astronomers and mathematicians. The earliest known Indian treatise containing references to these functions is Surya Siddhanta.

29Lecture No. 1

Ex.: Sinusoidal Signals

(a) Sinusoidal signal A cos(ωt + Φ) with phase Φ(b) Sinusoidal signal A sin (ωt + Φ) with phase Φ

30Lecture No. 1

Elementary Signals• Sinusoidal signals in discrete time

[ ] cos( )x n A n • Discrete-time sinusoidal signals may or may not be periodic.

• To be perodic,

[ ] cos( )2 radians

x n N A n NN m

31Lecture No. 1

Elementary Signals3. Complex Exponential Signals

cos sinje j

( )

Let

cos( ) sin( )cos( ) Re{ }sin( ) Im{ }

j

jwt j j t j t

j t

j t

B AeBe Ae e Ae

A t jA tA t BeA t Be

Given

32Lecture No. 1

Elementary Signals4. Exponentially Damped Sinusoidal Signals

( ) sin( ), 0tx t Ae t

Exponentially damped sinusoidal signal Ae-at

sin(ωt), with A = 60 and α = 6.

33Lecture No. 1

Elementary Signals5. Step function

1, 0[ ]

0, 0n

u nn

1, 0( )

0, 0t

u tt

34Lecture No. 1

Ex.: Step Function

Continuous-time version of the unit-step function of unit amplitude.

35Lecture No. 1

Elementary Signals6. Impulse Function or Dirac delta function

( ) 0 for 0

( ) 1

t t

t dt

1, 0

[ ] 0, 0

nn

n

It was introduced by theoretical physicist Paul Dirac. Dirac explicitly spoke of infinitely great values of his integrand. In the context of signal processing it is often referred to as the unit impulse symbol (or function). Its discrete analog is the Kronecker delta function which is usually defined on a finite domain and takes values 0 and 1.

physically represents an idealized point massor point charge

36Lecture No. 1

Ex.: Impulse Function

Discrete-time form of impulse.http://upload.wikimedia.org/wikipedia/commons/b/b4/Dirac_function_approximation.gif

The Dirac delta function as the limit (in the sense of distributions) of the sequence of zero-centerednormal distributions as

37Lecture No. 1

Elementary Signals7. Ramp Function

, 0( )

0, 0t t

r tt

, 0[ ]

0, 0n n

r nn

38Lecture No. 1

Ex.: Ramp FunctionRamp function of unit slope.

39Lecture No. 1

Systems• An interconnection of operations that transforms an

input signal into an output signal with properties different from those of the input signal.

• The signals may be of the continuous-time or discrete-time variety or a mixture of both.

• Let the overall operator H denote the action of a system.

( ) { ( )}[ ] { [ ]}

y t H x ty n H x n

- System Classification

40Lecture No. 1

Systems

Block diagram representation of operator H for (a) continuous time and (b) discrete time.

41

Systems

• Series Systems• Parallel Systems• Series-Parallel Systems• Feedback Systems

42Lecture No. 1

Properties of Systems• Stability

– A system is said to be bounded-input, bounded-output (BIBO) stable iff every bounded input results in a bounded output.

– The output of such a system does not diverge if the input does not diverge.

( ) , for all t

( ) , for all ty

x

y t M

x t M

- System Properties

43Lecture No. 1

Ex.: Stability of SystemDramatic photographs showing the collapse of the Tacoma Narrows suspension bridge on November 7, 1940. (a) Photograph showing the twisting motion of the bridge’s center span just before failure. (b) A few minutes after the first piece of concrete fell, this second photograph shows a 600-ft section of the bridge breaking out of the suspension span and turning upside down as it crashed in Puget Sound, Washington. Note the car in the top right-hand corner of the photograph.

44Lecture No. 1

Properties of Systems• Memory

– A system is said to posses memory if its output signal depends on past or future values of the input signal.

– A system is said to be memoryless if its output signal depends only on the present value of the input signal.

– Resister is memoryless since a voltage across it only depends on its input

– Capacitor has memory

1. : ( ) ( ( ) ( 2) ( 4))3

ex y t x t x t x t

45Lecture No. 1

Properties of Systems• Causality

– A system is said to be causal if the present value of the output signal depends only on the present or past values of the input signal.

– The output signal of a noncausal system depends on one or more future values of the input signal.

1. : ( ) ( ( ) ( 1) ( 2))3

.: ( ) ( ( 1) ( ) ( 1))

ex y t x t x t x t

ex y t x t x t x t

46Lecture No. 1

Properties of Systems• Invertibility

– A system is said to be invertible if the input of the system can be recovered from the output.

{ ( )} { { ( )}} { ( )} ( )inv inv invH y t H H x t H H x t x t

inv

I = identity operator

H inverse operator

invH H I

47Lecture No. 1

System Invertibility

The notion of system invertibility. The second operator Hinv is the inverse of the first operator H. Hence, the input x(t) is passed through the cascade correction of H and H-1

completely unchanged.

48Lecture No. 1

Properties of Systems• Time Invariance

– A system is said to be time invariant if a time delay or time advance of the input signal leads to an identical time shift in the output signal.

1 1

2 1 0

2 2 1 0

1 0 1 0

2 1 0

( ) s in ( ( ) )( ) s in ( ( ) ) ( ) ( )( ) s in ( ( ) ) s in ( ( ) )

( ) s in ( ( ) )( ) ( )

y t x ty t x ti f x t x t ty t x t x t t

y t t x t ty t y t t

49Lecture No. 1

Time Invariance

The notion of time invariance. (a) Time-shift operator St0

preceding operator H. (b) Time-shift operator St0 following operator H. These two situations are equivalent, provided that H is time invariant.

50Lecture No. 1

Properties of Systems• Linearity

– A system is said to be linear, if the following two properties are satisfied.

– Superposition

– Homogeneity

1 1

2 2

1 2 1 2

( ) ( ), then ( ) ( )( ) ( ), then ( ) ( )( ) ( ) ( ), then ( ) ( ) ( )

x t x t y t y tx t x t y t y tx t x t x t y t y t y t

1 1

1 1

( ) ( ), then ( ) ( )( ) ( ), then ( ) ( )

x t x t y t y tx t x t y t y t

51Lecture No. 1

Linearity

The linearity property of a system. (a) The combined operation of amplitude scaling and summation precedes the operator H for multiple inputs. (b) The operator H precedes amplitude scaling for each input; the resulting outputs are summed to produce the overall output y(t). If these two configurations produce the same output y(t), the operator H is linear.

52

Why Elementary Signals & Systems?

Lego Blocks

Systems Systems