1. 1 Introduction 1.1 What is a signal?
CHAPTER
Introduction
1.1 What is a signal?
A signal is formally defined as a function of one or more variables that conveys information on the nature of a physical phenomenon.
1.2 What is a system?
A system is formally defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals.
Figure 1.1 (p. 2) Block diagram representation of a system.
1.3 Overview of Specific Systems
★ Communication systems
Elements of a communication system
Fig. 1.2
1. Analog communication system: modulator + channel + demodulator
Signals_and_Systems_Simon Haykin & Barry Van Veen

2. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure 1.2 (p. 3) Elements of a communication system. The transmitter changes the message signal into a form suitable for transmission over the channel. The receiver processes the channel output (i.e., the received signal) to produce an estimate of the message signal.
◆ Modulation:
2. Digital communication system:
sampling + quantization + coding  transmitter  channel  receiver
◆ Two basic modes of communication:
Fig. 1.3
Broadcasting
Point-to-point communication
Radio, television
Telephone, deep-space communication
Signals_and_Systems_Simon Haykin & Barry Van Veen

3. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure 1.3 (p. 5) (a) Snapshot of Pathfinder exploring 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. (Courtesy of Jet Propulsion Laboratory.)
Signals_and_Systems_Simon Haykin & Barry Van Veen

4. Controller: digital computer1
CHAPTER
Introduction
★ Control systems
Figure 1.4 (p. 7) 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.
◆ Reasons for using control system: 1. Response, 2. Robustness
◆ Closed-loop control system: Fig. 1.4.
Controller: digital computer
(Fig. 1.5.)
Single-input, single-output (SISO) system
Multiple-input, multiple-output (MIMO) system
Signals_and_Systems_Simon Haykin & Barry Van Veen

5. Figure 1.5 (p. 8) NASA space shuttle launch. (Courtesy of NASA.)
CHAPTER
Introduction
Figure 1.5 (p. 8) NASA space shuttle launch. (Courtesy of NASA.)
Signals_and_Systems_Simon Haykin & Barry Van Veen

6. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
★ Microelectromechanical Systems (MEMS)
Structure of lateral capacitive accelerometers: Fig. 1-6 (a).
Figure 1.6a (p. 8) Structure of lateral capacitive accelerometers. (Taken from Yazdi et al., Proc. IEEE, 1998)
Signals_and_Systems_Simon Haykin & Barry Van Veen

7. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
SEM view of Analog Device’s ADXLO5 surface-micromachined polysilicon accelerometer: Fig. 1-6 (b).
Figure 1.6b (p. 9) SEM view of Analog Device’s ADXLO5 surface-micromachined polysilicon accelerometer. (Taken from Yazdi et al., Proc. IEEE, 1998)
Signals_and_Systems_Simon Haykin & Barry Van Veen

8. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
★ Remote Sensing
Remote sensing is defined as the process of acquiring information about an object of interest without being in physical contact with it
1. Acquisition of information = detecting and measuring the changes that the
object imposes on the field surrounding it.
2. Types of remote sensor:
Radar sensor
Infrared sensor
Visible and near-infrared sensor
X-ray sensor
※ Synthetic-aperture radar (SAR)
See Fig. 1.7
Satisfactory operation
High resolution
Ex. A stereo pair of SAR acquired from earth orbit with Shuttle Imaging Radar
(SIR-B)
Signals_and_Systems_Simon Haykin & Barry Van Veen

9. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure 1.7 (p. 11) Perspectival view of Mount Shasta (California), derived from a pair of stereo radar images acquired from orbit with the shuttle Imaging Radar (SIR-B). (Courtesy of Jet Propulsion Laboratory.)
Signals_and_Systems_Simon Haykin & Barry Van Veen

10. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
★ Biomedical Signal Processing
Morphological types of nerve cells: Fig. 1-8.
Figure 1.8 (p. 12) 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.)
Signals_and_Systems_Simon Haykin & Barry Van Veen

11. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
◆ Important examples of biological signal:
Figure (p. 13) 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.
Electrocardiogram (ECG)
Electroencephalogram (EEG)
Fig. 1-9
Signals_and_Systems_Simon Haykin & Barry Van Veen

12. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
★ Measurement artifacts:
1. Instrumental artifacts
2. Biological artifacts
3. Analysis artifacts
★ Auditory System
Figure (p. 14) (a) In this diagram, the basilar membrane in the cochlea is depicted as if it were uncoiled and stretched out flat; the “base” and “apex” refer to the cochlea, but the remarks “stiff region” and “flexible region” refer to the basilar membrane. (b) This diagram illustrates the traveling waves along the basilar membrane, showing their envelopes induced by incoming sound at three different frequencies.
Signals_and_Systems_Simon Haykin & Barry Van Veen

13. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
★ The ear has three main parts:
Outer ear: collection of sound
Middle ear: acoustic impedance match between the air and cochlear fluid
Conveying the variations of the tympanic membrane (eardrum)
3. Inner ear: mechanical variations → electrochemical or neural signal
★ Basilar membrane: Traveling wave
Fig
★ Analog Versus Digital Signal Processing
Digital approach has two advantages over analog approach:
Flexibility
Repeatability
1.4 Classification of Signals
Parentheses (‧)
1. Continuous-time and discrete-time signals
Continuous-time signals: x(t)
Fig
Discrete-time signals:
(1.1)
where t = nTs
Fig
Brackets [‧]
Signals_and_Systems_Simon Haykin & Barry Van Veen

14. Figure 1.11 (p. 17) Continuous-time signal.
CHAPTER
Introduction
Figure (p. 17) Continuous-time signal.
Figure (p. 17) (a) Continuous-time signal x(t). (b) Representation of x(t) as a discrete-time signal x[n].
Signals_and_Systems_Simon Haykin & Barry Van Veen

15. Symmetric about vertical axis Antisymmetric about origin1
CHAPTER
Introduction
Symmetric about vertical axis
2. Even and odd signals
Even signals:
(1.2)
Odd signals:
(1.3)
Example 1.1
Antisymmetric about origin
Consider the signal
Is the signal x(t) an even or an odd function of time?
<Sol.>
odd function
Signals_and_Systems_Simon Haykin & Barry Van Veen

16. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
◆ Even-odd decomposition of x(t):
Example 1.2
Find the even and odd components of the signal
where
<Sol.>
Even component:
(1.4)
(1.5)
Odd component:
Signals_and_Systems_Simon Haykin & Barry Van Veen

17. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
◆ Conjugate symmetric:
A complex-valued signal x(t) is said to be conjugate symmetric if
(1.6)
Refer to
Fig. 1-13
Problem 1-2
Let
3. Periodic and nonperiodic signals (Continuous-Time Case)
Periodic signals:
(1.7)
Figure (p. 20) (a) One example of continuous-time signal. (b) Another example of a continuous-time signal.
and
Fundamental frequency:
(1.8)
Angular frequency:
(1.9)
Signals_and_Systems_Simon Haykin & Barry Van Veen

18. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
◆ Example of periodic and nonperiodic signals: Fig
Figure (p. 21) (a) Square wave with amplitude A = 1 and period T = 0.2s. (b) Rectangular pulse of amplitude A and duration T1.
◆ Periodic and nonperiodic signals (Discrete-Time Case)
(1.10)
N = positive integer
Fundamental frequency of x[n]:
(1.11)
Signals_and_Systems_Simon Haykin & Barry Van Veen

19. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 21) Triangular wave alternative between –1 and +1 for Problem 1.3.
◆ Example of periodic and nonperiodic signals:
Fig and Fig
Figure (p. 22) Discrete-time square wave alternative between –1 and +1.
Signals_and_Systems_Simon Haykin & Barry Van Veen

20. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 22) Aperiodic discrete-time signal consisting of three nonzero samples.
4. Deterministic signals and random signals
A deterministic signal is a signal about which there is no uncertainty with respect to its value at any time.
Figure 1.13 ~ Figure 1.17
A random signal is a signal about which there is uncertainty before it occurs.
Figure 1.9
5. Energy signals and power signals
Instantaneous power:
If R = 1  and x(t) represents a current or a voltage, then the instantaneous power is
(1.12)
(1.14)
(1.13)
Signals_and_Systems_Simon Haykin & Barry Van Veen

21. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
The total energy of the continuous-time signal x(t) is
◆ Discrete-time case:
Total energy of x[n]:
(1.15)
(1.18)
Time-averaged, or average, power is
Average power of x[n]:
(1.16)
(1.19)
For periodic signal, the time-averaged power is
(1.17)
(1.20)
★ Energy signal:
If and only if the total energy of the signal satisfies the condition
★ Power signal:
If and only if the average power of the signal satisfies the condition
Signals_and_Systems_Simon Haykin & Barry Van Veen

22. Performed by amplifier1
CHAPTER
Introduction
1.5 Basic Operations on Signals
★ Operations Performed on dependent Variables
c = scaling factor
Amplitude scaling:
x(t)
(1.21)
Discrete-time case:
x[n]
Performed by amplifier
Addition:
(1.22)
Discrete-time case:
Multiplication:
Ex. AM modulation
(1.23)
Differentiation:
Figure (p. 26) Inductor with current i(t), inducing voltage v(t) across its terminals.
(1.24)
Inductor:
(1.25)
Integration:
(1.26)
Signals_and_Systems_Simon Haykin & Barry Van Veen

23. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Capacitor:
(1.27)
Figure (p. 27) Capacitor with voltage v(t) across its terminals, inducing current i(t).
★ Operations Performed on
independent Variables
Time scaling:
a >1  compressed
0 < a < 1  expanded
Fig
Figure (p. 27)
Time-scaling operation; (a) continuous-time signal x(t), (b) version of x(t) compressed by a factor of 2, and (c) version of x(t) expanded by a factor of 2.
Signals_and_Systems_Simon Haykin & Barry Van Veen

24. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Discrete-time case:
k = integer
Some values lost!
Figure (p. 28) Effect of time scaling on a discrete-time signal: (a) discrete-time signal x[n] and (b) version of x[n] compressed by a factor of 2, with some values of the original x[n] lost as a result of the compression.
Reflection:
The signal y(t) represents a reflected version of x(t) about t = 0.
Ex. 1-3
Consider the triangular pulse x(t) shown in Fig. 1-22(a). Find the reflected version of x(t) about the amplitude axis (i.e., the origin).
<Sol.> Fig.1-22(b).
Signals_and_Systems_Simon Haykin & Barry Van Veen

25. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 28) Operation of reflection: (a) continuous-time signal x(t) and (b) reflected version of x(t) about the origin.
t0 > 0  shift toward right
t0 < 0  shift toward left
Time shifting:
Ex. 1-4 Time Shifting: Fig
Figure (p. 29) Time-shifting operation: (a) continuous-time signal in the form of a rectangular pulse of amplitude 1.0 and duration 1.0, symmetric about the origin; and (b) time-shifted version of x(t) by 2 time shifts.
Signals_and_Systems_Simon Haykin & Barry Van Veen

26. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Discrete-time case:
where m is a positive or negative integer
★ Precedence Rule for Time Shifting and Time Scaling
1. Combination of time shifting and time scaling:
(1.28)
(1.29)
(1.30)
2. Operation order:
To achieve Eq. (1.28),
1st step: time shifting
2nd step: time scaling
Ex. 1-5 Precedence Rule for Continuous-Time Signal
Consider the rectangular pulse x(t) depicted in Fig. 1-24(a). Find y(t)=x(2t + 3).
<Sol.>
Case 1: Fig  Shifting first, then scaling
Case 2: Fig  Scaling first, then shifting
Signals_and_Systems_Simon Haykin & Barry Van Veen

27. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 31) The proper order in which the operations of time scaling and time shifting should be applied in the case of the continuous-time signal of Example 1.5. (a) Rectangular pulse x(t) of amplitude 1.0 and duration 2.0, symmetric about the origin. (b) Intermediate pulse v(t), representing a time-shifted version of x(t). (c) Desired signal y(t), resulting from the compression of v(t) by a factor of 2.
Signals_and_Systems_Simon Haykin & Barry Van Veen

28. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure 1.25 (p. 31) The incorrect way of applying the precedence rule. (a) Signal x(t). (b) Time-scaled signal v(t) = x(2t). (c) Signal y(t) obtained by shifting v(t) = x(2t) by 3 time units, which yields y(t) = x(2(t + 3)).
Ex. 1-6 Precedence Rule for Discrete-Time Signal
A discrete-time signal is defined by
Find y[n] = x[2x + 3].
Signals_and_Systems_Simon Haykin & Barry Van Veen

29. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
<Sol.> See Fig
Figure (p. 33) The proper order of applying the operations of time scaling and time shifting for the case of a discrete-time signal. (a) Discrete-time signal x[n], antisymmetric about the origin. (b) Intermediate signal v(n) obtained by shifting x[n] to the left by 3 samples. (c) Discrete-time signal y[n] resulting from the compression of v[n] by a factor of 2, as a result of which two samples of the original x[n], located at n = –2, +2, are lost.
Signals_and_Systems_Simon Haykin & Barry Van Veen

30. B and a are real parameters1
CHAPTER
Introduction
B and a are real parameters
1.6 Elementary Signals
★ Exponential Signals
(1.31)
Decaying exponential, for which a < 0
Growing exponential, for which a > 0
Figure (p. 34) (a) Decaying exponential form of continuous-time signal. (b) Growing exponential form of continuous-time signal.
Signals_and_Systems_Simon Haykin & Barry Van Veen

31. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Ex. Lossy capacitor: Fig
KVL Eq.:
(1.32)
RC = Time constant
(1.33)
Figure (p. 35) Lossy capacitor, with the loss represented by shunt resistance R.
Discrete-time case:
(1.34)
where
Fig. 1.30
★ Sinusoidal Signals
Fig. 1-31
◆ Continuous-time case:
(1.35)
where
periodicity
Signals_and_Systems_Simon Haykin & Barry Van Veen

32. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 35) (a) Decaying exponential form of discrete-time signal. (b) Growing exponential form of discrete-time signal.
Signals_and_Systems_Simon Haykin & Barry Van Veen

33. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 36) (a) Sinusoidal signal A cos( t + Φ) with phase Φ = +/6 radians. (b) Sinusoidal signal A sin ( t + Φ) with phase Φ = +/6 radians.
Signals_and_Systems_Simon Haykin & Barry Van Veen

34. Natural angular frequency of oscillation of the circuit1
CHAPTER
Introduction
Ex. Generation of a sinusoidal signal  Fig
Circuit Eq.:
(1.36)
(1.37)
where
Figure (p. 37) Parallel LC circuit, assuming that the inductor L and capacitor C are both ideal.
(1.38)
Natural angular frequency of oscillation of the circuit
◆ Discrete-time case :
(1.39)
Periodic condition:
(1.40) or
(1.41)
Ex. A discrete-time sinusoidal signal: A = 1,  = 0, and N = 12.
Fig
Signals_and_Systems_Simon Haykin & Barry Van Veen

35. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 38) Discrete-time sinusoidal signal.
Signals_and_Systems_Simon Haykin & Barry Van Veen

36. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Example 1.7 Discrete-Time Sinusoidal Signal
A pair of sinusoidal signals with a common angular frequency is defined by
and
Both x1[n] and x2[n] are periodic. Find their common fundamental period.
Express the composite sinusoidal signal
In the form y[n] = Acos(n + ), and evaluate the amplitude A and phase .
<Sol.>
(a) Angular frequency of both x1[n] and x2[n]:
This can be only for m = 5, 10, 15, …, which results in N = 2, 4, 6, …
(b) Trigonometric identity:
Let  = 5, then compare
x1[n] + x2[n] with the above equation to obtain that
Signals_and_Systems_Simon Haykin & Barry Van Veen

37. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
=   / 6
Accordingly, we may express y[n] as
★ Relation Between Sinusoidal and Complex Exponential Signals
1. Euler’s identity:
(1.41)
(1.42)
Complex exponential signal:
(1.35)
(1.42)
Signals_and_Systems_Simon Haykin & Barry Van Veen

38. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
◇ Continuous-time signal in terms of sine function:
(1.44)
(1.45)
2. Discrete-time case:
(1.46)
and
(1.47)
3. Two-dimensional representation of the complex
exponential e j  n for  = /4 and n = 0, 1, 2, …, 7.
: Fig
Projection on real axis: cos(n);
Projection on imaginary axis: sin(n)
Figure (p. 41) Complex plane, showing eight points uniformly distributed on the unit circle.
Signals_and_Systems_Simon Haykin & Barry Van Veen

39. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
★ Exponential Damped Sinusoidal Signals
(1.48)
Example for A = 60,
 = 6, and  = 0: Fig.1.35.
Figure (p. 41) Exponentially damped sinusoidal signal Ae at sin(t), with A = 60 and  = 6.
Signals_and_Systems_Simon Haykin & Barry Van Veen

40. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Ex. Generation of an exponential damped sinusoidal signal
 Fig
Circuit Eq.:
(1.49)
(1.50)
where
(1.51)
Figure (p. 42) Parallel LRC, circuit, with inductor L, capacitor C, and resistor R all assumed to be ideal.
Comparing Eq. (1.50) and (1.48), we have
◆ Discrete-time case:
(1.52)
x[n] n 1 2 3 4 1 2 3
★ Step Function
Figure (p. 43) Discrete-time version of step function of unit amplitude.
◆ Discrete-time case:
(1.53)
Fig
Signals_and_Systems_Simon Haykin & Barry Van Veen

41. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
◆ Continuous-time case:
Figure (p. 44) Continuous-time version of the unit-step function of unit amplitude.
(1.54)
Example 1.8 Rectangular Pulse
Consider the rectangular pulse x(t) shown in Fig (a). This pulse has an amplitude A and duration of 1 second. Express x(t) as a weighted sum of two step functions.
<Sol.>
1. Rectangular pulse x(t):
(1.55)
(1.56)
Example 1.9 RC Circuit
Find the response v(t) of RC circuit shown in Fig (a).
<Sol.>
Signals_and_Systems_Simon Haykin & Barry Van Veen

42. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 44) (a) Rectangular pulse x(t) of amplitude A and duration of 1 s, symmetric about the origin. (b) Representation of x(t) as the difference of two step functions of amplitude A, with one step function shifted to the left by ½ and the other shifted to the right by ½; the two shifted signals are denoted by x1(t) and x2(t), respectively. Note that x(t) = x1(t) – x2(t).
Signals_and_Systems_Simon Haykin & Barry Van Veen

43. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 45) (a) Series RC circuit with a switch that is closed at time t = 0, thereby energizing the voltage source. (b) Equivalent circuit, using a step function to replace the action of the switch.
1. Initial value:
2. Final value:
3. Complete solution:
(1.57)
Signals_and_Systems_Simon Haykin & Barry Van Veen

44. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
★ Impulse Function
Figure (p. 46) Discrete-time form of impulse.
◆ Discrete-time case:
(1.58)
Fig. 1.41
Figure (p. 46) Discrete-time form of impulse.
(t)
a(t)
Figure (p. 46) (a) Evolution of a rectangular pulse of unit area into an impulse of unit strength (i.e., unit impulse). (b) Graphical symbol for unit impulse. (c) Representation of an impulse of strength a that results from allowing the duration Δ of a rectangular pulse of area a to approach zero.
Signals_and_Systems_Simon Haykin & Barry Van Veen

45. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
◆ Continuous-time case:
Dirac delta function
(1.59)
(1.60)
1. As the duration decreases, the rectangular pulse approximates the impulse
more closely.
Fig
2. Mathematical relation between impulse and rectangular pulse function:
(1.61)
x(t): even function of t,  = duration.
x(t): Unit area.
Fig (a).
3. (t) is the derivative of u(t):
4. u(t) is the integral of (t):
(1.62)
(1.63)
Example 1.10 RC Circuit (Continued)
For the RC circuit shown in Fig (a), determine the current i (t) that flows through the capacitor for t  0.
Signals_and_Systems_Simon Haykin & Barry Van Veen

46. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
<Sol.>
Figure (p. 47) (a) Series circuit consisting of a capacitor, a dc voltage source, and a switch; the switch is closed at time t = 0. (b) Equivalent circuit, replacing the action of the switch with a step function u(t).
1. Voltage across the capacitor:
2. Current flowing through capacitor:
Signals_and_Systems_Simon Haykin & Barry Van Veen

47. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
◆ Properties of impulse function:
1. Even function:
(1.68)
(1.64)
2. Sifting property:
Ex. RLC circuit driven by impulsive
source: Fig
(1.65)
3. Time-scaling property:
For Fig (a), the voltage across the capacitor at time t = 0+ is
(1.66)
(1.69)
<p.f.>
Fig. 1.44
1. Rectangular pulse approximation:
(1.67)
2. Unit area pulse: Fig. 1.44(a).
Time scaling: Fig. 1.44(b).
Area = 1/a
Restoring unit area
ax(at)
Signals_and_Systems_Simon Haykin & Barry Van Veen

48. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 48) Steps involved in proving the time-scaling property of the unit impulse. (a) Rectangular pulse xΔ(t) of amplitude 1/Δ and duration Δ, symmetric about the origin. (b) Pulse xΔ(t) compressed by factor a. (c) Amplitude scaling of the compressed pulse, restoring it to unit area.
Figure (p. 49) (a) Parallel LRC circuit driven by an impulsive current signal. (b) Series LRC circuit driven by an impulsive voltage signal.
Signals_and_Systems_Simon Haykin & Barry Van Veen

49. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
★ Derivatives of The Impulse
Problem 1.24
1. Doublet:
(1.70)
2. Fundamental property of the doublet:
(1.71)
(1.72)
3. Second derivative of impulse:
(1.73)
★ Ramp Function
1. Continuous-time case:
(1.74) or
(1.75)
Fig. 1.46
Signals_and_Systems_Simon Haykin & Barry Van Veen

50. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
2. Discrete-time case:
Figure (p. 51) Ramp function of unit slope.
(1.76) or
(1.77)
Fig
x[n] n 1 2 3 4 1 2 3
Figure (p. 52) Discrete-time version of the ramp function.
Example 1.11 Parallel Circuit
Consider the parallel circuit of Fig (a) involving a dc current source I0 and an initially uncharged capacitor C.
The switch across the capacitor is suddenly opened at time t = 0. Determine the current i(t) flowing through the capacitor and the voltage v(t) across it for t  0.
<Sol.>
1. Capacitor current:
Signals_and_Systems_Simon Haykin & Barry Van Veen

51. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
2. Capacitor voltage:
Figure (p. 52) (a) Parallel circuit consisting of a current source, switch, and capacitor, the capacitor is initially assumed to be uncharged, and the switch is opened at time t = 0. (b) Equivalent circuit replacing the action of opening the switch with the step function u(t).
Signals_and_Systems_Simon Haykin & Barry Van Veen

52. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
1.7 Systems Viewed as Interconnections of Operations
A system may be viewed as an interconnection of operations that transforms an input signal into an output signal with properties different from those of the input signal.
1. Continuous-time case:
(1.78)
2. Discrete-time case:
(1.79)
Figure (p. 53) Block diagram representation of operator H for (a) continuous time and (b) discrete time.
Fig (a) and (b).
Example 1.12 Moving-average system
Consider a discrete-time system whose output signal y[n] is the average of the three most recent values of the input signal x[n], that is
Formulate the operator H for this system; hence, develop a block diagram representation for it.
Signals_and_Systems_Simon Haykin & Barry Van Veen

53. Both Mx and My represent some finite positive number1
CHAPTER
Introduction
<Sol.>
1. Discrete-time-shift operator Sk: Fig
Shifts the input x[n] by k time units to produce an output equal to x[n  k].
Figure (p. 54) Discrete-time-shift operator Sk, operating on the discrete-time signal x[n] to produce x[n – k].
2. Overall operator H for the moving-average
system:
Fig
Fig (a): cascade form; Fig (b): parallel
form.
1.8 Properties of Systems
★ Stability
1. A system is said to be bounded-input, bounded-output (BIBO) stable if and
only if every bounded input results in a bounded output.
2. The operator H is BIBO stable if the output signal y(t) satisfies the condition
(1.80)
Both Mx and My represent some finite positive number
whenever the input signals x(t) satisfy the condition
(1.81)
Signals_and_Systems_Simon Haykin & Barry Van Veen

54. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 54) Two different (but equivalent) implementations of the moving-average system: (a) cascade form of implementation and (b) parallel form of implementation.
Signals_and_Systems_Simon Haykin & Barry Van Veen

55. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
One famous example of an unstable system:
Figure 1.52a (p. 56) Dramatic photographs showing the collapse of the Tacoma Narrows suspension bridge on November 7, (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. (Courtesy of the Smithsonian Institution.)
Signals_and_Systems_Simon Haykin & Barry Van Veen

56. The moving-average system is stable.1
CHAPTER
Introduction
Example 1.13 Moving-average system (continued)
Show that the moving-average system described in Example 1.12 is BIBO stable.
<p.f.>
1. Assume that:
2. Input-output relation:
The moving-average system is stable.
Signals_and_Systems_Simon Haykin & Barry Van Veen

57. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Example 1.14 Unstable system
Consider a discrete-time system whose input-output relation is defined by
where r > 1. Show that this system is unstable.
<p.f.>
1. Assume that:
2. We find that
With r > 1, the multiplying factor rn diverges for increasing n.
The system is unstable.
★ Memory
A system is said to possess memory if its output signal depends on past or future values of the input signal.
A system is said to possess memoryless if its output signal depends only on the present values of the input signal.
Signals_and_Systems_Simon Haykin & Barry Van Veen

58. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Ex.: Resistor
Memoryless !
Ex.: Inductor
Memory !
Ex.: Moving-average system
Memory !
Ex.: A system described by the input-output relation
Memoryless !
★ Causality
A system is said to be causal if its present value of the output signal depends only on the present or past values of the input signal.
A system is said to be noncausal if its output signal depends on one or more future values of the input signal.
Signals_and_Systems_Simon Haykin & Barry Van Veen

59. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Ex.: Moving-average system
Causal !
Ex.: Moving-average system
Noncausal !
 A causal system must be capable of operating in real time.
★ Invertibility
A system is said to be invertible if the input of the system can be recovered from the output.
Figure (p. 59) The notion of system invertibility. The second operator H inv is the inverse of the first operator H. Hence, the input x(t) is passed through the cascade correction of H and H inv completely unchanged.
1. Continuous-time system: Fig
x(t) = input; y(t) = output
H = first system operator;
H inv = second system operator
Signals_and_Systems_Simon Haykin & Barry Van Veen

60. H inv = inverse operator1
CHAPTER
Introduction
2. Output of the second system:
H inv = inverse operator
3. Condition for invertible system:
I = identity operator
(1.82)
Example 1.15 Inverse of System
Consider the time-shift system described by the input-output relation
where the operator S t0 represents a time shift of t0 seconds. Find the inverse of this system.
<Sol.>
1. Inverse operator S t 0:
2. Invertibility condition:
Time shift of t0
Signals_and_Systems_Simon Haykin & Barry Van Veen

61. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Example 1.16 Non-Invertible System
Show that a square-law system described by the input-output relation
is not invertible.
<p.f.>
Since the distinct inputs x(t) and  x(t) produce the same output y(t). Accordingly, the square-law system is not invertible.
★ Time Invariance
A system is said to be time invariance if a time delay or time advance of the input signal leads to an identical time shift in the output signal.
 A time-invariant system do not change with time.
Figure 1.55 (p.61) 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.
Signals_and_Systems_Simon Haykin & Barry Van Veen

62. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
1. Continuous-time system:
2. Input signal x1(t) is shifted in time by t0 seconds:
S t 0 = operator of a time shift equal to t0
3. Output of system H:
(1.83)
4. For Fig (b), the output of system H is y1(t  t0):
(1.84)
5. Condition for time-invariant system:
(1.85)
Signals_and_Systems_Simon Haykin & Barry Van Veen

63. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
x1(t) = v(t)
y1(t) = i(t)
Example 1.17 Inductor
The inductor shown in figure is described by the input-output relation:
where L is the inductance. Show that the inductor so described is time invariant.
<Sol.>
1. Let
x1(t)
x1(t  t0)
Response y2(t) of the inductor to
x1(t  t0) is
(A)
2. Let y1(t  t0) = the original output of the inductor, shifted by t0 seconds:
(B)
3. Changing variables:
(A)
Inductor is time invariant.
Signals_and_Systems_Simon Haykin & Barry Van Veen

64. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
x1(t) = v(t)
y1(t) = i(t)
Example 1.18 Thermistor
Let R(t) denote the resistance of the thermistor, expressed as a function of time. We may express the input-output relation of the device as
Show that the thermistor so described is time variant.
<Sol.>
1. Let response y2(t) of the thermistor to
x1(t  t0) is
2. Let y1(t  t0) = the original output of the thermistor due to x1(t), shifted by t0
seconds:
3. Since R(t)  R(t  t0)
Time variant!
Signals_and_Systems_Simon Haykin & Barry Van Veen

65. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
★ Linearity
A system is said to be linear in terms of the system input (excitation) x(t) and the system output (response) y(t) if it satisfies the following two properties of superposition and homogeneity:
1. Superposition:
2. Homogeneity:
a = constant factor
 Linearity of continuous-time system
1. Operator H represent the continuous-tome system.
2. Input:
x1(t), x2(t), …, xN(t)  input signal; a1, a2, …, aN  Corresponding weighted factor
(1.86)
3. Output:
(1.87)
Signals_and_Systems_Simon Haykin & Barry Van Veen

66. Superposition and homogeneity1
CHAPTER
Introduction
Superposition and homogeneity
(1.88)
where
(1.89)
4. Commutation and Linearity:
(1.90)
Fig. 1.56
 Linearity of discrete-time system
Same results, see Example 1.19.
Example 1.19 Linear Discrete-Time system
Consider a discrete-time system described by the input-output relation
Show that this system is linear.
Signals_and_Systems_Simon Haykin & Barry Van Veen

67. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 64) 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.
<p.f.>
1. Input:
2. Resulting output signal:
Signals_and_Systems_Simon Haykin & Barry Van Veen

68. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
where
Linear system!
Example 1.20 Nonlinear Continuous-Time System
Consider a continuous-time system described by the input-output relation
Show that this system is nonlinear.
<p.f.>
1. Input:
2. Output:
Here we cannot write
Nonlinear system!
Signals_and_Systems_Simon Haykin & Barry Van Veen

69. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Example 1.21 Impulse Response of RC Circuit
For the RC circuit shown in Fig. 1.57, determine the impulse response y(t).
<Sol.>
1. Recall: Unit step response
Figure (p. 66) RC circuit for Example 1.20, in which we are given the capacitor voltage y(t) in response to the step input x(t) = y(t) and the requirement is to find y(t) in response to the unit-impulse input x(t) = (t).
(1.91)
2. Rectangular pulse input: Fig
x(t) = x(t)
1/
/2
/2
Figure (p. 66) Rectangular pulse of unit area, which, in the limit, approaches a unit impulse as Δ0.
3. Response to the step
functions x1(t) and x2(t):
Signals_and_Systems_Simon Haykin & Barry Van Veen

70. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Next, recognizing that
i) (t) = the limiting form of the pulse x(t):
(1.92)
Signals_and_Systems_Simon Haykin & Barry Van Veen

71. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
ii) The derivative of a continuous function of time, say, z(t):
(1.92)
Cancel each other!
(1.93)
Signals_and_Systems_Simon Haykin & Barry Van Veen

72. 1 Introduction 1.9 Noise Noise  Unwanted signals
CHAPTER
Introduction
1.9 Noise
Noise  Unwanted signals
1. External sources of noise: atmospheric noise, galactic noise, and human-
made noise.
2. Internal sources of noise: spontaneous fluctuations of the current or voltage
signal in electrical circuit. (electrical noise)
Fig
★ Thermal Noise
Thermal noise arises from the random motion of electrons in a conductor.
Two characteristics of thermal noise:
1. Time-averaged value:
2T = total observation interval of noise
(1.94)
As T  ,
Refer to Fig
2. Time-average-squared value:
k = Boltzmann’s constant = 1.38  10 23 J/K
Tabs = absolute temperature
(1.95)
As T  ,
(1.96)
Signals_and_Systems_Simon Haykin & Barry Van Veen

73. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure 1.60
(p. 68) Sample waveform of electrical noise generated by a thermionic diode with a heated cathode. Note that the time-averaged value of the noise voltage displayed is approximately zero.
Signals_and_Systems_Simon Haykin & Barry Van Veen

74. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
 Thevenin’s equvalent circuit: Fig. 1.61(a), Norton’s equivalent circuit:
Fig. 1.61(b).
Noise voltage generator:
Noise current generator:
(1.97)
where G = 1/R = conductance [S].
 Maximum power transfer
theorem: the maximum
possible power is transferred
from a source of internal
resistance R to a load of
resistance Rl when R = Rl.
Figure (p. 70) (a) Thévenin equivalent circuit of a noisy resistor. (b) Norton equivalent circuit of the same resistor.
Under matched condition, the available power is
Signals_and_Systems_Simon Haykin & Barry Van Veen

75. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
 Two operating factor that affect available noise power:
1. The temperature at which the resistor is maintained.
2. The width of the frequency band over which the noise voltage across the
resistor is measured.
★ Other Sources of Electrical Noise
1. Shot noise: the discrete nature of current flow electronic devices
2. Ex. Photodetector:
1) Electrons are emitted at random times, k, where  < k < 
2) Total current flowing through photodetector:
(1.98)
where
is the current pulse generated at time k.
3. 1/f noise: The electrical noise whose time-averaged power at a given
frequency is inversely proportional to the frequency.
Signals_and_Systems_Simon Haykin & Barry Van Veen

76. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
1.10 Theme Example
★ Differentiation and Integration: RC Circuits
1. Differentiator  Sharpening of a pulse
differentiator
x(t)
y(t)
(1.99)
1) Simple RC circuit: Fig
2) Input-output relation:
(1.100)
Figure (p. 71) Simple RC circuit with small time constant, used as an approximator to a differentiator.
If RC (time constant) is small enough such that (1.100) is dominated by the second term v2(t)/RC, then
(1.101)
Signals_and_Systems_Simon Haykin & Barry Van Veen

77. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Input: x(t) = RCv1(t); output: y(t) = v2(t)
2. Integrator  smoothing of an input signal
integrator
x(t)
y(t)
(1.102)
1) Simple RC circuit: Fig
2) Input-output relation:
(1.103)
If RC (time constant) is large enough such that (1.103) is dominated by the first term RCv2(t), then
Figure (p. 72) Simple RC circuit with large time constant used as an approximator to an integrator.
Input: x(t) = [1/(RC)v1(t)]; output: y(t) = v2(t)
Signals_and_Systems_Simon Haykin & Barry Van Veen

78. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
★ MEMS Accelerometer
1. Model: second-order mass-damper-spring system
Fig
M = proof mass, K = effective spring constant, D = damping factor, x(t) =external acceleration, y(t) =displacement of proof mass, Md 2y(t)/dt 2 = inertial force of proof mass, Ddy(t)/dt = damping force, Ky(t) = spring force.
2. Force Eq.:
Figure (p. 73) Mechanical lumped model of an accelerometer.
(1.105)
1) Natural frequency:
2) Quality factor:
[rad/sec]
(1.106)
(1.107)
Signals_and_Systems_Simon Haykin & Barry Van Veen

79. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
(1.105)
(1.108)
★ Radar Range Measurement
1. A periodic sequence of radio frequency (RF) pulse: Fig
T0 = duration [sec], 1/T = repeated frequency, fc = RF frequency [MHz~GHz]
Figure (p. 74) Periodic train of rectangular FR pulses used for measuring molar ranges.
 The sinusoidal signal acts as a carrier.
Signals_and_Systems_Simon Haykin & Barry Van Veen

80. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
2. Round-trip time = the time taken by a radar pulse to reach the target and for
the echo from the target to come back to the radar.
(1.109)
d = radar target range, c = light speed.
3. Two issues of concern in range measurement:
1) Range resolution: The duration T0 of the pulse places a lower limit on the
shortest round-trip delay time that the radar can measure.
Smallest target range: dmin = cT0/2 [m]
2) Range ambiguity: The interpulse period T places an upper limit on the
largest range that the radar can measure.
Largest target range: dmax = cT/2 [m]
★ Moving-Average Systems
1. N-point moving-average system:
Fig
(1.110)
x(t) = input signal
The value N determines the degree to which the system smooths the input data.
Signals_and_Systems_Simon Haykin & Barry Van Veen

81. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure 1.66a
(p. 75) (a) Fluctuations in the closing stock price of Intel over a three-year period.
Signals_and_Systems_Simon Haykin & Barry Van Veen

82. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure 1.66b
(p. 76) (b) Output of a four-point moving-average system.
N = 4 case
Signals_and_Systems_Simon Haykin & Barry Van Veen

83. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure 1.66c
(p. 76) (c) Output of an eight-point moving-average system.
N = 8 case
Signals_and_Systems_Simon Haykin & Barry Van Veen

84. Tdiff = smallest time difference between different path1
CHAPTER
Introduction
2. For a general moving-average system, unequal weighting is applied to past
values of the input:
(1.111)
★ Multipath Communication Channels
1. Channel noise degrades the performance of a communication system.
2. Another source of degradation of channel: dispersive nature, i.e., the channel
has memory.
3. For wireless system, the dispersive characteristics result from multipath
propagation.
Fig
 For a digital communication, multipath propagation manifests itself in the
form of intersymbol interference (ISI).
4. Baseband model for multipath propagation: Tapped-delay line  Fig
Tdiff = smallest time difference between different path
(1.112)
Signals_and_Systems_Simon Haykin & Barry Van Veen

85. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 77) Example of multiple propagation paths in a wireless communication environment.
Signals_and_Systems_Simon Haykin & Barry Van Veen

86. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 78) Tapped-delay-line model of a linear communication channel, assumed to be time-invariant.
Signals_and_Systems_Simon Haykin & Barry Van Veen

87. Linearly weighted moving-average system1
CHAPTER
Introduction
5. PTdiff = the longest time delay of any significant path relative to the arrival of
the signal.
The coefficients wi are used to approximate the gain of each path.
For P =1, then
0x(t) = direct path, 1x(t Tdiff) = single reflected path
6. Discrete-time case:
Linearly weighted moving-average system
(1.113)
The term recursive signifies the dependence of the output signal on its own past values.
For P =1, then
(1.114)
★ Recursive Discrete-Time Computations
1. First-order recursive discrete-time filter: Fig
(1.115)
x[n] = input, y[n] = output
where  is a constant.
Signals_and_Systems_Simon Haykin & Barry Van Veen

88. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 79) Block diagram of first-order recursive discrete-time filter. The operator S shifts the output signal y[n] by one sampling interval, producing y[n – 1]. The feedback coefficient  determines the stability of the filter.
 Fig. 1.69: linear discrete-time feedback system.
2. Solution of Eq.(1.115):
(1.116)
r  
(1.117)
Setting k  1 = l, Eq.(1.117) becomes
(1.118)
Signals_and_Systems_Simon Haykin & Barry Van Veen

89. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
3. Three special cases (depending on ):
Accumulator
1)  = 1:
(1.116)
(1.119) 2)
Leaky accumulator
Stable in BIBO sense 3)
Amplified accumulator
Unsatble in BIBO sense
1.11 Exploring Concepts with MATLAB
 MATLAB Signal Processing Toolbox
1. Time vector: Sampling interval Ts of 1 ms on the interval from 0 to 1 s
t = 0:.001:1;
2. Vector n:
n = 0:1000;
★ Periodic Signals
1. Square wave: A =amplitude, w0 = fundamental frequency, rho = duty cycle
A*square(w0*t, rho);
Signals_and_Systems_Simon Haykin & Barry Van Veen

90. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Ex. Obtain the square wave shown in Fig (a) by using MATLAB.
<Sol.>
>> A = 1;
>> w0 =10*pi;
>> rho = 0.5;
>> t = 0:.001:1;
>> sq = A*square(w0*t, rho);
>> plot (t, sq)
>> axis([ ])
2. Triangular wave: A =amplitude, w0 = fundamental frequency, w = width
A*sawtooth(w0*t, w);
Ex. Obtain the triangular wave shown in Fig by using MATLAB.
<Sol.>
>> A = 1;
>> w0 =10*pi;
>> w = 0.5;
>> t = 0:0.001:1;
>> tri = A*sawtooth(w0*t, w);
>> plot (t, tri)
Signals_and_Systems_Simon Haykin & Barry Van Veen

91. 1 Introduction 3. Discrete-time signal: stem(n, x);
CHAPTER
Introduction
3. Discrete-time signal:
stem(n, x);
x = vector, n = discrete time vector
Ex. Obtain the discrete-time square wave shown in Fig by using MATLAB.
<Sol.>
>> A = 1;
>> omega =pi/4;
>> n = -10:10;
>> x = A*square(omega*n);
>> stem(n, x)
4. Decaying exponential:
B*exp(-a*t);
Growing exponential:
B*exp(a*t);
Ex. Obtain the decaying exponential signal shown in Fig (a) by using
MATLAB.
<Sol.>
>> B = 5;
>> a = 6;
>> t = 0:.001:1;
>> x = B*exp(-a*t); % decaying exponential
>> plot (t, x)
Signals_and_Systems_Simon Haykin & Barry Van Veen

92. 1
CHAPTER
Introduction
Ex. Obtain the growing exponential signal shown in Fig (b) by using
MATLAB.
<Sol.>
>> B = 1;
>> a = 5;
>> t = 0:0.001:1;
>> x = B*exp(a*t); % growing exponential
>> plot (t, x)
Ex. Obtain the decaying exponential sequence shown in Fig (a) by using
MATLAB.
<Sol.>
>> B = 1;
>> r = 0.85;
>> n = -10:10;
>> x = B*r.^n; % decaying exponential sequence
>> stem (n, x)
★ Sinusoidal Signals
A = amplitude, w0 = frequency, phi = phase angle
1. Cosine signal:
2. Sine signal:
A*cos(w0*t + phi);
A*sin(w0*t + phi);
Signals_and_Systems_Simon Haykin & Barry Van Veen

93. 1
CHAPTER
Introduction
Ex. Obtain the sinusoidal signal shown in Fig (a) by using MATLAB.
<Sol.>
>> A = 4;
>> w0 =20*pi;
>> phi = pi/6;
>> t = 0:.001:1;
>> cosine = A*cos(w0*t + phi);
>> plot (t, cosine)
Ex. Obtain the discrete-time sinusoidal signal shown in Fig by using
MATLAB.
<Sol.>
>> A = 1;
>> omega =2*pi/12; % angular frequency
>> n = -10:10;
>> y = A*cos(omega*n);
>> stem (n, y)
.*  element-by-element multiplication
★ Exponential Damped Sinusoidal Signals
1. Exponentially damped sinusoidal signal:
MATLAB Format: A*sin(w0*t + phi).*exp(-a*t);
Signals_and_Systems_Simon Haykin & Barry Van Veen

94. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Ex. Obtain the waveform shown in Fig by using MATLAB.
<Sol.>
>> A = 60;
>> w0 =20*pi;
>> phi = 0;
>> a = 6;
>> t = 0:.001:1;
>> expsin = A*sin(w0*t + phi).*exp(-a*t);
>> plot (t, expsin)
Ex. Obtain the exponentially damped sinusoidal sequence shown in Fig by
using MATLAB.
<Sol.>
>> A = 1;
>> omega =2*pi/12; % angular frequency
>> n = -10:10;
>> y = A*cos(omega*n);
>> r = 0.85;
>> x = A*r.^n; % decaying exponential sequence
>> z = x.*y; % elementwise multiplication
>> stem (n, z)
Signals_and_Systems_Simon Haykin & Barry Van Veen

95. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Figure (p. 84) Exponentially damped sinusoidal sequence.
Signals_and_Systems_Simon Haykin & Barry Van Veen

96. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
★ Step, Impulse, and Ramp Functions
Ex. Generate a rectangular pulse centered at origin on the interval [-1, 1].
◆ MATLAB command:
1. M-by-N matrix of ones: ones (M, N)
2. M-by-N matrix of zeros: zeros (M, N)
<Sol.>
 Unit amplitude step function:
>> t = -1:1/500:1;
>> u1 = [zeros(1, 250), ones(1, 751)];
>> u2 = [zeros(1, 751), ones(1, 250)];
>> u = u1 – u2;
u = [zeros(1, 50), ones(1, 50)];
 Discrete-time impulse:
delta = [zeros(1, 49), 1, zeros(1, 49)];
 Ramp sequence:
ramp = 0:.1:10
★ User-Defined Functions
1. Two types M-files exist: scripts and functions.
Scripts, or script files automate long sequences of commands; functions, or function files, provide extensibility to MATLAB by allowing us to add new functions.
2. Procedure for establishing a function M-file:
Signals_and_Systems_Simon Haykin & Barry Van Veen

97. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
1) It begins with a statement defining the function name, its input arguments, and
its output arguments.
2) It also includes additional statements that compute the values to be returned.
3) The inputs may be scalars, vectors, or matrices.
Ex. Obtain the rectangular pulse depicted in Fig (a) with the use of an
M-file.
The function size returns a two-element vector containing the row and column dimensions of a matrix.
The function find returns the indices of a vector or matrix that satisfy a prescribed relation
<Sol.>
>> function g = rect(x)
>> g = zeros(size(x));
>> set1 = find(abs(x)<= 0.5);
>> g(set1) = ones(size(set1));
3. The new function rect.m can be used
liked any other MATLAB function.
Ex. find(abs(x)<= T) returns the indices
of the vector x, where the absolute
value of x is less than or equal to T.
Ex. To generate a rectangular pulse:
>> t = -1:1/500:1;
>> plot(t, rect(t));
Signals_and_Systems_Simon Haykin & Barry Van Veen

98. Signals_and_Systems_Simon Haykin & Barry Van Veen1
CHAPTER
Introduction
Signals_and_Systems_Simon Haykin & Barry Van Veen