Introduction
CHAPTER
★ 1.8.6 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:
x (t )  x1 (t )
y (t )  y1 (t )
x (t )  x1 (t )  x2 (t )
x ( t )  x2 ( t )
y (t )  y2 (t )
y (t )  y1 (t )  y2 (t )
2. Homogeneity:
y (t )
x(t )
ay (t )
ax(t )
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 
N
Corresponding weighted factor
x(t )   ai xi (t ) (1.86)
i 1
3. Output:
N
y (t )  H {x(t )}  H { ai xi (t )}
i 1
(1.87)
Introduction
CHAPTER
N
y (t )   ai yi (t )
(1.88)
i 1
Superposition and
homogeneity
where
yi (t )  H {xi (t )}, i  1, 2, ..., N .
(1.89)
4. Commutation and Linearity:
N
y (t )  H { ai xi (t )}
i 1
N
  ai H {xi (t )}
(1.90)
Fig. 1.56
i 1
N
  ai yi (t )
i 1
 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
y[n]  nx[n]
Show that this system is linear.
Introduction
CHAPTER

Figure 1.56 (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:
N
x[n]   ai xi [n]
i 1
2. Resulting output signal:
Introduction
CHAPTER
N
N
N
y[n]  n ai xi [n]   ai nxi [n]   ai yi [n]
i 1
i 1
where
yi [n]  nxi [n]
i 1
Linear system!
Example 1.20 Nonlinear Continuous-Time System
Consider a continuous-time system described by the input-output relation
y(t )  x(t ) x(t  1)
Show that this system is nonlinear.
<p.f.>
N
1. Input: x(t ) 
ai xi (t )

i 1
2. Output:
N
N
N
N
i 1
j 1
i 1 j 1
y(t )   ai xi (t ) a j x j (t  1)  ai a j xi (t ) x j (t  1)
Here we cannot write
y (t )   i 1 ai yi (t )
N
Nonlinear system!
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
y(t )  (1  e t / RC )u(t ), x(t )  u(t )
(1.91)
2. Rectangular pulse input: Fig. 1.58.
x(t) = x(t)
1

u (t  )

2
1

x2 (t )  u (t  )

2
x1 (t ) 
1/
Figure 1.57 (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).
Figure 1.58 (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):
/2
/2
Introduction
CHAPTER
 


 t   /( RC )  
1

y1  1  e  2 
u
t

 
,
 
2
 
x(t )  x1 (t )
 t   /( RC ) 
1
 
 2
y2  1  e
 u  t  ,
 
2
 
x ( t )  x2 ( t )


Next, recognizing that
x (t )  x1 (t )  x2 (t )
1
1
(1  e ( t  / 2) /( RC )  )u(t   / 2)  (1  e ( t  / 2) /( RC )  )u( t   / 2)


1
1
 (u(t   / 2)  u(t   / 2))  ( e ( t  / 2) /( RC ) u( t   / 2)  e ( t  / 2) /( RC )  )u( t   / 2))


y (t ) 
i) (t) = the limiting form of the pulse x(t):
 (t )  lim x (t )
0
(1.92)
Introduction
CHAPTER
ii) The derivative of a continuous function of time, say, z(t):
d
1


z (t )  lim{ ( z (t  )  z (t  ))}
0 
dt
2
2
y (t )  lim y (t )
0
d  t /( RC )
  (t )  (e
u (t ))
dt
d t /( RC )
 t /( RC ) d
  (t )  e
u (t )  u (t ) (e
)
dt
dt
1 t /( RC )
 t /( RC )
  (t )  e
 (t ) 
e
u (t ), x(t )   (t )
RC
(1.92)
Cancel each other!
y (t ) 
1  t /( RC )
e
u(t ), x(t )   (t )
RC
(1.93)
Introduction
CHAPTER
1.9 Noise
Noise  Unwanted signals
1. External sources of noise: atmospheric noise, galactic noise, and humanmade noise.
2. Internal sources of noise: spontaneous fluctuations of the current or voltage
signal in electrical circuit. (electrical noise)
Fig. 1.60.
★ 1.9.1 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
v  lim
T  2T

T
T
v(t )dt
(1.94)
As T  , v  0
2. Time-average-squared value:
1
v  lim
T  2T
2
As T  ,

T
T
v 2 (t )dt
k = Boltzmann’s constant =
1.38  10 23 J/K
Tabs = absolute temperature
(1.95)
v 2  4kTabs Rf
Refer to Fig. 1.60.
volts2
(1.96)
CHAPTER
Introduction
Figure 1.60
(p. 68)
Sample waveform
of electrical noise
generated by a
thermionic diode
with a heated
cathode. Note
that the timeaveraged value of
the noise voltage
displayed is
approximately
zero.
CHAPTER
Introduction
 Thevenin’s equvalent circuit: Fig. 1.61(a), Norton’s equivalent circuit:
Fig. 1.61(b).
Noise voltage generator:
v(t )  v 2
Noise current generator:
1 T v(t ) 2
i  lim
(
) dt
T  2T T
R
(1.97)
 4kTabsGf amps2
2
where G = 1/R = conductance [S].
 Maximum power transfer
Figure 1.61 (p. 70)
theorem: the maximum
possible power is transferred (a) Thévenin equivalent circuit of a noisy resistor.
(b) Norton equivalent circuit of the same resistor.
from a source of internal
resistance R to a load of
resistance Rl when R = Rl.
Under matched condition, the available power is
kTabs f watts
Introduction
CHAPTER
 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.
★ 1.9.2 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:
x(t ) 

 h(t  
k 
k
)
(1.98)
where h(t   k ) 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.
Introduction
CHAPTER
1.10 Theme Example
★ 1.10.1 Differentiation and Integration: RC Circuits
1. Differentiator  Sharpening of a pulse
x(t)
d
y (t )  x(t )
dt
y(t)
differentiator
(1.99)
1) Simple RC circuit: Fig. 1.62.
2) Input-output relation:
d
1
d
v2 (t ) 
v2 (t )  v1 (t ) (1.100)
dt
RC
dt
If RC (time constant) is small enough
such that (1.100) is dominated by the
second term v2(t)/RC, then
1
d
v2 (t )  v1 (t )
RC
dt
v2 (t )  RC
Figure 1.62 (p. 71)
Simple RC circuit with small time
constant, used as an approximator
to a differentiator.
d
v1 (t ) for RC small
dt
(1.101)
Introduction
CHAPTER
Input: x(t) = RCv1(t); output: y(t) = v2(t)
2. Integrator  smoothing of an input signal
x(t)
t
y(t ) 
x( )d
(1.102)
y(t)

integrator

1) Simple RC circuit: Fig. 1.63.
2) Input-output relation:
d
RC v2 (t )  v2 (t )  v1 (t )
dt
t
t


RCv2 (t )   v2 ( )d   v1 ( )d
(1.103)
If RC (time constant) is large enough
such that (1.103) is dominated by the
first term RCv2(t), then
t
RCv2 (t )   v1 ( )d
Figure 1.63 (p. 72)
Simple RC circuit with large time
constant used as an approximator
to an integrator.

1 t
v2 ( t ) 
v1 ( )d


RC
for large RC
Input: x(t) = [1/(RC)v1(t)];
output: y(t) = v2(t)
Introduction
CHAPTER
★ 1.10.2 MEMS Accelerometer
1. Model: second-order mass-damper-spring system
Fig. 1.64.
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 1.64 (p. 73)
Mechanical lumped model
of an accelerometer.
d 2 y (t )
dy (t )
Mx(t )  M
D
 Ky (t )
2
dt
dt
d 2 y (t ) D dy (t ) K


y (t )  x(t )
2
dt
M dt
M
1) Natural frequency:
n 
K
M
[rad/sec]
(1.105)
2) Quality factor:
(1.106)
Q
KM
D
(1.107)
CHAPTER
(1.105)
Introduction
d 2 y(t ) n dy(t )
2



n y (t )  x(t )
dt 2
Q dt
(1.108)
★ 1.10.3 Radar Range Measurement
1. A periodic sequence of radio frequency (RF) pulse: Fig. 1.65.
T0 = duration [sec], 1/T = repeated frequency, fc = RF frequency [MHz~GHz]
Figure 1.65 (p. 74)
Periodic train of rectangular FR pulses used for measuring molar ranges.
 The sinusoidal signal acts as a carrier.
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.
2d

(1.109)
c
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]
★ 1.10.4 Moving-Average Systems
1. N-point moving-average system:
Fig. 1.66.
1
y[n] 
N
N 1
 x[n  k ]
(1.110)
x(t) = input signal
k 0
The value N determines the degree to which the system smooths the input data.
CHAPTER
Introduction
Figure 1.66a
(p. 75)
(a) Fluctuations in
the closing stock
price of Intel over
a three-year
period.
CHAPTER
Introduction
Figure 1.66b
(p. 76)
(b) Output of a
four-point
moving-average
system.
N = 4 case
CHAPTER
Introduction
Figure 1.66c
(p. 76)
(c) Output of an
eight-point
moving-average
system.
N = 8 case
Introduction
CHAPTER
2. For a general moving-average system, unequal weighting is applied to past
values of the input:
N 1
y[n]   ak x[n  k ]
(1.111)
k 0
★ 1.10.5 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. 1.67.
 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. 1.68.
p
y (t )   i x(t  iTdiff )
i 0
(1.112)
Tdiff = smallest time difference
between different path
CHAPTER
Introduction
Figure 1.67 (p. 77)
Example of multiple propagation paths in a wireless communication environment.
CHAPTER
Introduction
Figure 1.68 (p. 78)
Tapped-delay-line model of a linear communication channel, assumed to be timeinvariant.
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
y(t )  0 x(t )  1x(t  Tdiff )
0x(t) = direct path, 1x(t Tdiff) = single reflected path
6. Discrete-time case:
Linearly weighted
moving-average system
p
y[n]   k x[n  k ]
(1.113)
k 0
For P =1, then
y[n]  x[n]  ax[n  1]
(1.114)
★ 1.10.6 Recursive Discrete-Time Computations
The term recursive
signifies the dependence
of the output signal on its
own past values.
1. First-order recursive discrete-time filter: Fig. 1.69.
y[n]  x[n]   y[n  1]
where  is a constant.
(1.115)
x[n] = input, y[n] = output
Introduction
CHAPTER
Figure 1.69 (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):

y[n]    k x[n  k ]
r
(1.116)
k 0

y[n]  x[n]    k x[n  k ]
(1.117)
k 1
Setting k  1 = l, Eq.(1.117) becomes


y[n]  x[n]    [n  1  l ] x[n]     l [n  1  l ]
l 0
1 l
l 0
(1.118)
Introduction
CHAPTER
y[n]  x[n]   y[n  1]
3. Three special cases (depending on ):
1)  = 1:
(1.116)
Accumulator

y[n]   x[n  k ]
(1.119)
k 0
2)   1: Leaky accumulator
Stable in BIBO sense
3)   1: Amplified accumulator
1.11 Exploring Concepts with MATLAB
 MATLAB Signal Processing Toolbox
Unsatble in
BIBO sense
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;
★ 1.11.1 Periodic Signals
1. Square wave: A =amplitude, w0 = fundamental frequency, rho = duty cycle
A*square(w0*t, rho);
CHAPTER
Introduction
Ex. Obtain the square wave shown in Fig. 1.14 (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([0 1 -1.1 1.1])
2. Triangular wave: A =amplitude, w0 = fundamental frequency, w = width
A*sawtooth(w0*t, w);
Ex. Obtain the triangular wave shown in Fig. 1.15 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)
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. 1.16 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. 1.28 (a) by using
MATLAB.
<Sol.> >> B = 5;
>> a = 6;
>> t = 0:.001:1;
>> x = B*exp(-a*t); % decaying exponential
>> plot (t, x)
CHAPTER
Introduction
Ex. Obtain the growing exponential signal shown in Fig. 1.28 (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. 1.30 (a) by using
MATLAB.
<Sol.>
>> B = 1;
>> r = 0.85;
>> n = -10:10;
>> x = B*r.^n; % decaying exponential sequence
>> stem (n, x)
★ 1.11.3 Sinusoidal Signals
1. Cosine signal:
A*cos(w0*t + phi);
2. Sine signal:
A*sin(w0*t + phi);
A = amplitude, w0 =
frequency, phi =
phase angle
CHAPTER
Introduction
Ex. Obtain the sinusoidal signal shown in Fig. 1.31 (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. 1.33 by using
MATLAB.
<Sol.>
>> A = 1;
>> omega =2*pi/12; % angular frequency
.*  element-by-element
>> n = -10:10;
multiplication
>> y = A*cos(omega*n);
>> stem (n, y)
★ 1.11.4 Exponential Damped Sinusoidal Signals
1. Exponentially damped sinusoidal signal:
x(t )  A sin(0t   )e  at
MATLAB Format: A*sin(w0*t + phi).*exp(-a*t);
CHAPTER
Introduction
Ex. Obtain the waveform shown in Fig. 1.35 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. 1.70 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)
CHAPTER
Introduction
Figure 1.70 (p. 84)
Exponentially damped sinusoidal sequence.
CHAPTER
Introduction
★ 1.11.5 Step, Impulse, and Ramp Functions
◆ MATLAB command:
1. M-by-N matrix of ones: ones (M, N)
2. M-by-N matrix of zeros: zeros (M, N)
Ex. Generate a rectangular pulse
centered at origin on the interval
[-1, 1].
<Sol.>
 Unit amplitude step function:
u = [zeros(1, 50), ones(1, 50)];
 Discrete-time impulse:
delta = [zeros(1, 49), 1, zeros(1, 49)];
>> t = -1:1/500:1;
>> u1 = [zeros(1, 250), ones(1, 751)];
>> u2 = [zeros(1, 751), ones(1, 250)];
>> u = u1 – u2;
 Ramp sequence:
ramp = 0:.1:10
★ 1.11.6 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:
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. 1.39 (a) with the use of an
M-file.
1. The function size returns a two<Sol.>
element vector containing the row
>> function g = rect(x)
and column dimensions of a matrix.
>> g = zeros(size(x));
2. The function find returns the
>> set1 = find(abs(x)<= 0.5);
indices of a vector or matrix that
>> g(set1) = ones(size(set1));
satisfy a prescribed relation
3. The new function rect.m can be used
liked any other MATLAB function.
Ex. find(abs(x)<= T) returns the indices
Ex. To generate a rectangular pulse:
of the vector x, where the absolute
>> t = -1:1/500:1;
value of x is less than or equal to T.
>> plot(t, rect(t));