Introduction CHAPTER ★ 1.8.6 Linearity A system is said to be linear in terms of the system input...

IntroductionIntroduction CHAPTER

★ ★ 1.8.6 Linearity1.8.6 LinearityA 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:

1( ) ( )x t x t 1( ) ( )y t y t

2( ) ( )x t x t 2( ) ( )y t y t1 2( ) ( ) ( )x t x t x t

1 2( ) ( ) ( )y t y t y t 2. Homogeneity:

( )x t ( )y t ( )ax t ( )ay ta = constant

factor

Linearity of continuous-time system1. Operator H represent the continuous-tome system.2. Input:

1

( ) ( )N

i ii

x t a x t

(1.86)

x1(t), x2(t), …, xN(t) input signal; a1, a2, …, aN

Corresponding weighted factor

3. Output:

1

( ) { ( )} { ( )}N

i ii

y t H x t H a x t

(1.87)


1

( ) ( )N

i ii

y t a y t

(1.88) Superposition and

homogeneity

where

( ) { ( )}, 1, 2, ..., .i iy t H x t i N (1.89)

4. Commutation and Linearity:

1

1

1

( ) { ( )}

{ ( )}

( )

N

i ii

N

i ii

N

i ii

y t H a x t

a H x t

a y t

(1.90) Fig. 1.56Fig. 1.56

Linearity of discrete-time system Same results, see Example 1.19Example 1.19.Example 1.19 Linear Discrete-Time systemConsider a discrete-time system described by the input-output relation

[ ] [ ]y n nx nShow that this system is linear.


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.><p.f.>1. Input:

1

[ ] [ ]N

i ii

x n a x n

2. Resulting output signal:


1 1 1

[ ] [ ] [ ] [ ]N N N

i i i i i ii i i

y n n a x n a nx n a y n

where [ ] [ ]i iy n nx n

Linear system!

Example 1.20 Nonlinear Continuous-Time SystemConsider a continuous-time system described by the input-output relation

Show that this system is nonlinear.<p.f.><p.f.>

( ) ( ) ( 1)y t x t x t

1. Input:1

( ) ( )N

i ii

x t a x t

2. Output:

1 1 1 1

( ) ( ) ( 1) ( ) ( 1)N N N N

i i j j i j i ji j i j

y t a x t a x t a a x t x t

Here we cannot write 1

( ) ( )N

i iiy t a y t

Nonlinear system!

1/

/2 /2


Example 1.21 Impulse Response of RC CircuitFor the RC circuit shown in Fig. 1.57Fig. 1.57, determine the impulse response y(t).

Figure 1.57 (p. 66)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).

<Sol.><Sol.>1. Recall: Unit step response

/( ) (1 ) ( ), ( ) ( )t RCy t e u t x t u t (1.91)

2. Rectangular pulse input: Fig. 1.58Fig. 1.58.

Figure 1.58 (p. 66)Rectangular pulse of unit area, which, in the limit, approaches a unit impulse as Δ0.

x(t) = x(t)

1

1( ) ( )

2x t u t

2

1( ) ( )

2x t u t

3. Response to the step

functions x1(t) and x2(t):


/( )2

1 1

11 , ( ) ( )

2

t RC

y e u t x t x t

/( )2

2 2

11 , ( ) ( )

2

t RC

y e u t x t x t

Next, recognizing that

1 2( ) ( ) ( )x t x t x t

( / 2) /( ) ( / 2) /( )

( / 2) /( ) ( / 2) /( )

1 1( ) (1 ) ( / 2) (1 ) ( / 2)

1 1( ( / 2) ( / 2)) ( ( / 2) ) ( / 2))

t RC t RC

t RC t RC

y t e u t e u t

u t u t e u t e u t

(1.92) i) (t) = the limiting form of the pulse x(t):

0( ) lim ( )t x t


ii) The derivative of a continuous function of time, say, z(t):

0

1( ) lim{ ( ( ) ( ))}

2 2

dz t z t z t

dt

(1.92)

0

/( )

/( ) /( )

/( ) /( )

( ) lim ( )

( ) ( ( ))

( ) ( ) ( ) ( )

1 ( ) ( ) ( ), ( ) ( )

t RC

t RC t RC

t RC t RC

y t y t

dt e u t

dtd d

t e u t u t edt dt

t e t e u t x t tRC

/( )1( ) ( ), ( ) ( )t RCy t e u t x t t

RC (1.93)

Cancel each other!


1.9 Noise1.9 NoiseNoise Unwanted signals1. 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. 1.60.Fig. 1.60.

★ ★ 1.9.1 Thermal Noise1.9.1 Thermal NoiseThermal noise arises from the random motion of electrons in a conductor.

Two characteristics of thermal noise: 1. Time-averaged value:

1lim ( )

2

T

TTv v t dt

T (1.94)

2T = total observation interval of noise

As T , 0v Refer to Fig. 1.60.Refer to Fig. 1.60.

2. Time-average-squared value:

2 21lim ( )

2

T

TTv v t dt

T (1.95)

As T , 2 24 voltsabsv kT R f (1.96)

k = Boltzmann’s constant = 1.38 10 23 J/K

Tabs = absolute temperature


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.


Thevenin’s equvalent circuit: Fig. 1.61(a),Fig. 1.61(a), Norton’s equivalent circuit: Fig. 1.61(b).Fig. 1.61(b).

Figure 1.61 (p. 70)(a) Thévenin equivalent circuit of a noisy resistor. (b) Norton equivalent circuit of the same resistor.

Noise voltage generator:2( )v t v

Noise current generator:

2 2

2

1 ( )lim ( )

2

4 amps

T

TT

abs

v ti dt

T R

kT G f

(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.

Under matched condition, the available power iswattsabskT f


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 Noise1.9.2 Other Sources of Electrical Noise1. Shot noise: the discrete nature of current flow electronic devices2. Ex. Photodetector:

1) Electrons are emitted at random times, k, where < k < 2) Total current flowing through photodetector:

( ) ( )kk

x t h t

(1.98)

( )kh t 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.


1.10 Theme Example1.10 Theme Example

★ ★ 1.10.1 Differentiation and Integration: 1.10.1 Differentiation and Integration: RCRC Circuits Circuits 1. Differentiator Sharpening of a pulse

differentiatorx(t) y(t)

( ) ( )d

y t x tdt

1) Simple RC circuit: Fig. 1.62Fig. 1.62.

Figure 1.62 (p. 71)Simple RC circuit with small time constant, used as an approximator to a differentiator.

2) Input-output relation:

(1.99)

2 2 1

1( ) ( ) ( )

d dv t v t v t

dt RC dt (1.100)

If RC (time constant) is small enough such that (1.100) is dominated by the second term v2(t)/RC, then

2 1

1( ) ( )

dv t v t

RC dt 2 1( ) ( ) for small

dv t RC v t RC

dt (1.101)


Input: x(t) = RCv1(t); output: y(t) = v2(t)

2. Integrator smoothing of an input signal

1) Simple RC circuit: Fig. 1.63Fig. 1.63.

integratorx(t) y(t)

( ) ( )t

y t x d

(1.102)

2) Input-output relation:

2 2 1( ) ( ) ( )t t

RCv t v d v d

2 2 1( ) ( ) ( )d

RC v t v t v tdt

(1.103)

If RC (time constant) is large enough such that (1.103) is dominated by the first term RCv2(t), then

2 1( ) ( )t

RCv t v d

2 1

1( ) ( ) for large

tv t v d RC

RC

Figure 1.63 (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)


★ ★ 1.10.2 MEMS Accelerometer1.10.2 MEMS Accelerometer1. Model: second-order mass-damper-spring system

Figure 1.64 (p. 73)Mechanical lumped model of an accelerometer.

Fig. 1.64.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 for

ce of proof mass, Ddy(t)/dt = damping force, Ky(t) = spring force.

2. Force Eq.:2

2

( ) ( )( ) ( )

d y t dy tMx t M D Ky t

dt dt

2

2

( ) ( )( ) ( )

d y t D dy t Ky t x t

dt M dt M (1.105)

1) Natural frequency:

n

K

M (1.106) [rad/sec]

2) Quality factor:

KMQ

D (1.107)


(1.105) 2

22

( ) ( )( ) ( )n

n

d y t dy ty t x t

dt Q dt

(1.108)

★ ★ 1.10.3 Radar Range Measurement1.10.3 Radar Range Measurement1. A periodic sequence of radio frequency (RF) pulse: Fig. 1.65Fig. 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.


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

c (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: ddminmin = = cTcT00/2/2 [m]2) Range ambiguity: The interpulse period T places an upper limit on the largest range that the radar can measure.

Largest target range: ddmaxmax = = cTcT/2/2 [m]

★ ★ 1.10.4 Moving-Average Systems1.10.4 Moving-Average Systems1. N-point moving-average system:

1

0

1[ ] [ ]

N

k

y n x n kN

(1.110) x(t) = input signal

The value N determines the degree to which the system smooths the input data.

Fig. 1.66.Fig. 1.66.


Figure 1.66a (p. 75)(a) Fluctuations in the closing stock price of Intel over a three-year period.


Figure 1.66b (p. 76)(b) Output of a four-point moving-average system.N = 4 case


Figure 1.66c (p. 76)(c) Output of an eight-point moving-average system.

N = 8 case


2. For a general moving-average system, unequal weighting is applied to past values of the input:

1

0

[ ] [ ]N

kk

y n a x n k

(1.111)

★ ★ 1.10.5 Multipath Communication Channels1.10.5 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.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. 1.68Fig. 1.68.

0

( ) ( )p

i diffi

y t x t iT

(1.112) Tdiff = smallest time difference bet

ween different path


Figure 1.67 (p. 77)Example of multiple propagation paths in a wireless communication environment.


Figure 1.68 (p. 78)Tapped-delay-line model of a linear communication channel, assumed to be time-invariant.


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

0 1( ) ( ) ( )diffy t x t x t T

0x(t) = direct path, 1x(t Tdiff) = single reflected path

6. Discrete-time case:

0

[ ] [ ]p

kk

y n x n k

(1.113)

For P =1, then

[ ] [ ] [ 1]y n x n ax n (1.114)

Linearly weighted moving-average system

★ ★ 1.10.6 Recursive Discrete-Time Computations1.10.6 Recursive Discrete-Time Computations

1. First-order recursive discrete-time filter: Fig. 1.69Fig. 1.69.

The term recursive signifies the dependence of the output signal on its

own past values.

x[n] = input, y[n] = output[ ] [ ] [ 1]y n x n y n (1.115)

where is a constant.


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):

1

[ ] [ ] [ ]k

k

y n x n x n k

(1.116) 0

[ ] [ ]k

k

y n x n k

(1.117)

Setting k 1 = l, Eq.(1.117) becomes

1

0 0

[ ] [ ] [ 1 ] [ ] [ 1 ]l l

l l

y n x n n l x n n l

(1.118)

r


[ ] [ ] [ 1]y n x n y n 3. Three special cases (depending on ):

1) = 1:

0

[ ] [ ]k

y n x n k

(1.116) (1.119)

Accumulator

2) 1: Leaky accumulator

3) 1: Amplified accumulator

1.11 Exploring Concepts with MATLAB1.11 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;

★ ★ 1.11.1 Periodic Signals 1.11.1 Periodic Signals 1. Square wave: A =amplitude, w0 = fundamental frequency, rho = duty cycle

A*square(w0*t, rho);

Stable in BIBO senseStable in BIBO sense

Unsatble in BIUnsatble in BIBO senseBO sense


Ex. Obtain the square wave shown in Fig. 1.14 (a)Fig. 1.14 (a) by using MATLAB.

<Sol.><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. 1.15Fig. 1.15 by using MATLAB.

>> A = 1;>> w0 =10*pi;>> w = 0.5;>> t = 0:0.001:1;>> tri = A*sawtooth(w0*t, w);>> plot (t, tri)

<Sol.><Sol.>


3. Discrete-time signal: stem(n, x);x = vector, n = discrete time vector

Ex. Obtain the discrete-time square wave shown in Fig. 1.16Fig. 1.16 by using MATLAB.

>> A = 1;>> omega =pi/4;>> n = -10:10;>> x = A*square(omega*n);>> stem(n, x)

<Sol.><Sol.>

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)Fig. 1.28 (a) by using MATLAB.

>> B = 5;>> a = 6;>> t = 0:.001:1;>> x = B*exp(-a*t); % decaying exponential>> plot (t, x)

<Sol.><Sol.>


Ex. Obtain the growing exponential signal shown in Fig. 1.28 (b)Fig. 1.28 (b) by using MATLAB.

>> B = 1;>> a = 5;>> t = 0:0.001:1;>> x = B*exp(a*t); % growing exponential>> plot (t, x)

<Sol.><Sol.>

Ex. Obtain the decaying exponential sequence shown in Fig. 1.30 (a)Fig. 1.30 (a) by using MATLAB.<Sol.><Sol.>

>> B = 1;>> r = 0.85;>> n = -10:10;>> x = B*r.^n; % decaying exponential sequence>> stem (n, x)

★ ★ 1.11.3 Sinusoidal Signals 1.11.3 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


Ex. Obtain the sinusoidal signal shown in Fig. 1.31 (a)Fig. 1.31 (a) by using MATLAB.

>> A = 4;>> w0 =20*pi;>> phi = pi/6;>> t = 0:.001:1;>> cosine = A*cos(w0*t + phi);>> plot (t, cosine)

<Sol.><Sol.>

Ex. Obtain the discrete-time sinusoidal signal shown in Fig. 1.33Fig. 1.33 by using MATLAB.

>> A = 1;>> omega =2*pi/12; % angular frequency>> n = -10:10;>> y = A*cos(omega*n);>> stem (n, y)

<Sol.><Sol.>

★ ★ 1.11.4 Exponential Damped Sinusoidal Signals 1.11.4 Exponential Damped Sinusoidal Signals

1. Exponentially damped sinusoidal signal:

0( ) sin( ) atx t A t e MATLAB Format:MATLAB Format: A*sin(w0*t + phi).*exp(-a*t);

.*.* element-by-element multiplication


Ex. Obtain the waveform shown in Fig. 1.35Fig. 1.35 by using MATLAB.

<Sol.><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. 1.70Fig. 1.70 by using MATLAB.<Sol.><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)


Figure 1.70 (p. 84)Exponentially damped sinusoidal sequence.


★ ★ 1.11.5 Step, Impulse, and Ramp Functions 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)

Unit amplitude step function:

u = [zeros(1, 50), ones(1, 50)];

Discrete-time impulse:

delta = [zeros(1, 49), 1, zeros(1, 49)];

Ramp sequence:

ramp = 0:.1:10

Ex.Ex. Generate a rectangular pulse centered at origin on the interval [-1, 1].

<Sol.><Sol.>

>> t = -1:1/500:1;>> u1 = [zeros(1, 250), ones(1, 751)];>> u2 = [zeros(1, 751), ones(1, 250)];>> u = u1 – u2;

★ ★ 1.11.6 User-Defined Functions 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:


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)Fig. 1.39 (a) with the use of an M-file.<Sol.><Sol.>

>> function g = rect(x)>> g = zeros(size(x));>> set1 = find(abs(x)<= 0.5);>> g(set1) = ones(size(set1));

1. The function sizesize returns a two-element vector containing the row and column dimensions of a matrix.

2. The function findfind returns the indices of a vector or matrix that satisfy a prescribed relation

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.

3. The new function rect.m can be used liked any other MATLAB function.

>> t = -1:1/500:1;>> plot(t, rect(t));

Ex. To generate a rectangular pulse:

Introduction CHAPTER ★ 1.8.6 Linearity A system is said to be linear in terms of the system input...

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