Summer 2008
Signals & Systems
S.F. Hsieh
I. Introduction to Signals and Systems
1
Goal of Signals & Systems
to develop mathematical models/techniques for continuous-/discrete-time signal and system analysis/synthesis(design).
• Signals(functions) are variables(time/space) that carry information,
Ex. f (t) speech(voltage/current picked up from microphone, a computer file xxx.wav, stock
market index, ECG(ElectroCardioGram); f (x, y)/f (x, y) image xxx.gif or xxx.jpg; f (t, x, y)
video xxx.mpg.
• Systems(mapping) process input signals to produce output signals. They can be hardwares(DSP processors/circuits/automobile, economic system) or softwares(programs).
2
Classifications of signals
1. Digital/analog: the amplitude is discrete(quantized) or not.
2. Continuous-/Discrete-time: (MATLAB commands: plot, stem)
• Continuous-time signal: x(t), −∞ < t < ∞, where the time-variable t is continuous.
• Discrete-time signal: x[n] ≡ x(nT ), n = 0, ±1, ±2, . . ., where T represents the sampling
time. Note that x[3 12 ] is NOT defined. Many discrete-time systems are computer programs.
x(t)
1
0.8
0.6
0.4
0.2
0
2
4
6
8
10
12
14
16
2
4
6
8
x[n]
0.8
0.6
0.4
0.2
0
−8
−6
−4
−2
I-1
0
3. Deterministic/Random:
• Deterministic signals:
– periodic(Fourier series)
x(t) = x(t + T ), ∀t
minimum T is the fundamental period.
– aperiodic(Fourier transform)
• Random signals(such as noise): probability density function, mean, autocorrelation, power
spectral density.
4. Energy/Power signals:
• Size of a signal x(t):
– energy:
E≡
Z
+∞
−∞
|x(t)|2 dt
if x(t) is the voltage applied across a dummy load of one-Ω resistor, then x2 (t) is the
power assumption (Watts). Integration of power over time is the total energy of the
signal, namely, E indicates the energy that can be extracted from the signal.
∗ Continuous-time: T
R
Total energy over the time interval t1 ≤ t ≤ t2 is tt12 |x(t)|2 dt.
P 2
∗ Discrete-time: total energy over the time interval n1 ≤ n ≤ n2 is nn=n
|x[n]|2 .
1
– average power:
Z
1 T /2
|x(t)|2 dt
P ≡ lim
T →∞ T −T /2
∗ If x(t) is periodic, then its average power becomes
Z
1 T /2
|x(t)|2 dt
P =
T −T /2
√
∗ The rms(root-mean-square) value of x(t) is P .
• Definitions:
– x(t) is an energy signal if it has a finite energy Ex < ∞ and its power Px = 0.
– x(t) is a power signal if it has a finite power Px < ∞, and energy Ex = ∞.
• (Lathi Ex 1.2, p 72)
2
– Power of a single real sinusoid x(t) = C cos(ω0 t + θ) is Px = C2 , where C is a positive
amplitude.
– Power of a single exponential y(t) = Dejω0 t is Py = |D|2 , where D is a complex number
comprised of amplitude and phase.
P
– Power of sum of orthogonal complex exponentials z(t) = i Di ejωi t with distinct
P
P
frequencies ωi ’s is equal to the sum of individual powers Pz = i Pi = i |Di |2 .
– (Q) Suppose z(t) = x(t) + y(t). Under what condition will the power of z(t) be equal
to the sum of those of x(t) and y(t), i.e., Pz = Px + Py ?
• Examples:
(a)
(b)
(c)
(d)
(e)
x(t) = e−3t u(t) is an energy signal with energy 1/6.
x(t) = Aej2παt is a power signal with power A2 .
A rectangular pulse A ⊓ (t/τ ) of amplitude A and duration τ has energy A2 τ .
A ⊓ (t/τ ) · cos(ω0 t + θ) has energy A2 τ /2.
P
t−nT0
2
A rectangular pulse train ∞
n:−∞ ⊓( τ ), T0 > τ , is a power signal with P = A τ /T0
I-2
(f) A ramp signal f (t) = t and an everlasting exponential g(t) = e−at are neither energy
nor power signals.
(g) Is x(t) = 10 a power or energy signal?
(h) What is the energy of a sinc function sinc(x) = sinπxπx ? (Parseval’s thm in Chp 7’s
Fourier transform can solve it easily.)
3
Basic operations on signals
1. Amplitude:
(a) amplitude-scale: x(t) =⇒ ax(t), where a > 0,
• a > 1: amplify(gain > 1)
• 0 < a < 1: attenuate(gain < 1)
(b) negation: x(t) =⇒ −x(t), change of polarity
(c) level-shift: x(t) =⇒ x(t) + c
• c > 0: shift up by c
• c < 0: shift down by c
(d) Note that sum of two lines is itself a line, too.
x3 (t) = x1 (t) + x2 (t)
x1 (t)
XXX
XXX x2 (t)
XXX
-
t
[Ex] Given x(t), plot y(t) = −2x(t) + 3.
(“multiply/divide first, then add/subtract”)
-1
0
-1
x(t)
1
0
A
A
- t
1
-2
A
A
3
1
-
t
1
@
@
−2x(t)
-
-1
2. change of Time-variable: Assume that x(t) looks like
1
-1
x(t)
-
1
t
I-3
0
1
t
(a) time-scale: x(t) =⇒ x(αt), where α > 0
• α > 1, time-compression:
1
− 21
x(2t)
-
1
2
• α < 1, time-expansion:
x( 2t )
1
-
-2
t
2
t
This may seem to hurt your instinct. For justification, Let y(t) = x(2t). Choose a few t’s
and plot y(t), we have y(0) = x(0)(still centered around the origin), y( 12 ) = x(2 21 ) = x(1),
and y(− 12 ) = x(−2 12 ) = x(−1).
(b) time-reversal(inversion): x(t) =⇒ x(−t)
1
A
x(−t)
A
A
AA
t
-1
1
The right-hand-side is mirrored about the origin, and vice versa.
(c) time-shift: x(t) =⇒ x(t + β)
• β > 0, left-shifted(advanced)
-
−β
• β < 0, right-shifted(delayed):
0
x(t + β)
0
t
x(t − β)
-
β
t
Again, this seems to be unbelievable. A mindful reader will not hesitate to play the same
game: Let y(t) = x(t − 3). y(3) = x(3 − 3) = x(0), y(0) = x(0 − 3) = x(−3), etc. From
which, she/he convinces herself/himself.
(d) Combined operation: x(t) =⇒ x(αt + β) ,
[method 1]:
• Let y(t) = x(t + β), i.e., advance/delay x(t) by β to obtain y(t).
• Let z(t) = y(αt), i.e., compress/expand y(t) by α.
As a check, z(t) = y(αt) = x(αt + β). The above procedure can change its order. BUT, be
careful!
[method 2]:
I-4
• Let v(t) = x(αt), i.e., compress/expand y(t) by α.
• Let w(t) = v(t + αβ ), INSTEAD OF w′ (t) = v(t + β).
As a check, w(t) = v(t + αβ ) = x(α[t + αβ ]) = x(αt + β) = z(t), while w′ (t) = v(t + β) =
x(α[t + β]) = x(αt + αβ) 6= z(t)!
Note:
• if you have difficulty in combining w(t) and v(t), you can use more dummy variables
to avoid confusion:
β
v(s) = x(αs), w(t) = v(t + )
α
so that
β
w(t) = v(s)|s=t+ β = x(α[t + ]) = x(αt + β)
α
α
• [Method 1] is preferred due to its algebraic simplicity.
[Ex] Given x(t), plot y(t) = x(2t − 3), and z(t) = x(−2t + 3).
(“subtract/add first, then divide/multiply”, a rule which is exactly opposite to the one used in
the amplitude operations.)
It is strongly recommended to check y(t)/z(t) at some t′ s where x(t) changes abruptly(and t = 0).
• Given x(t)
1
x(t)
-
-1
1
• Right shift by 3, x(t − 3)
1
x(t − 3)
-
3
2
t
4
t
• Compress by 2, x(2t − 3)
1
-
2
t
• (Check)
let x(2t − 3) = x(0),
let x(2t − 3) = x(−1),
let x(2t − 3) = x(2),
we have t = 3/2
we have t = 1
we have t = 5/2
Similarly,
• Given x(t)
1
-1
x(t)
-
1
t
I-5
indeed y(3/2) = x(2t − 3) = x(0)
indeed y(1) = x(2t − 3) = x(−1)
indeed y(5/2) = x(2t − 3) = x(2)
• Left shift by 3, x(t + 3)
1
-4
x(t + 3)
-
t
-3 -2
• Compress by 2 and time-reversal(whichever comes first), x(−2t + 3)
E
1
E
E
EE
-
2
t
• (Check)
let x(−2t + 3) = x(0),
let x(−2t + 3) = x(−1),
let x(−2t + 3) = x(2),
4
we have t = 3/2
we have t = 2
we have t = 1/2
indeed y(3/2) = x(−2t + 3) = x(0)
indeed y(2) = x(−2t + 3) = x(−1)
indeed y(1/2) = x(−2t + 3) = x(2)
Signal characteristics
1. Even/Odd
• Even function(signal): x(t) = x(−t), ∀t, symmetric about the origin,
#c
#
c
#
c
• Odd function(signal): x(t) = −x(−t), ∀t, anti-symmetric about the origin,
6
b
b
b
Q
Q
Q
Q
Q
b
b
• (Fact) Every signal x(t) can be decomposed as a sum of even and odd signals: x(t) =
and xo (t) ≡ Od{x(t} ≡ x(t)−x(−t)
.
xe (t) + xo (t), where xe (t) ≡ Ev{x(t} ≡ x(t)+x(−t)
2
2
2
A
A
A
A
x(t)
1
=
6
x (t)
l e
l
l
2. Periodic signals: x(t) = x(t + nT ), ∀t. T is its period.
+
6
xo (t)
@
@
@
@
Sum of several continuous-time periodic signals may not be period, depending on the relationship
among their fundamental periods.
3. Causal signal: x(t) = 0, ∀t < 0; anticausal signal: x(t) = 0, ∀t > 0.
I-6
5
Basic Signal Models
1. Complex exponential signals:
x(t) = Cest ≡ Aejφ e(σ+jω)t · · · · · · s ≡ σ + jω in Laplace transform
x[n] = Cz n ≡ Aejφ (rejω )n · · · · · · z ≡ rejω
in Z transform
(a) ω = 0 and φ = 0, x(t) = Aeσt is real exponential(non-oscillating),
i. σ > 0 or r > 1: exponentially growing,
ii. σ < 0 or r < 1: exponentially decaying,
iii. σ = 0 or r = 0: constant.
(b) ω 6= 0,
i. σ = 0 or r = 1: x(t) = Aejφ ejωt = A cos(ωt + φ) + jA sin(ωt + φ) or x[n] = Aejφ ejωn
is a non-damping cisoid(complex sinusoid). · · · · · · will see in Fourier transforms.
ii. σ > 0 or r > 1: x(t) = Aeσt ej(ωt+φ) is an exponentially increasing(unstable) cisoid
iii. σ < 0 or r < 1: x(t) = Aeσt ej(ωt+φ) is an exponentially damped(decreasing) cisoid
[Euler’s formula]
ejωt ≡ cos ωt + j sin ωt and e−jωt ≡ cos ωt − j sin ωt
1 jωt
1
cos ωt =
(e + e−jωt ) and sin ωt = (ejωt − e−jωt )
2
2j
10
5
damping factor =0.5
frequency=2.7pi
0
−5
−10
−15
−5
t=−5:0.01:5;
x=exp(0.5*t).*cos(2.7*pi*t);
subplot(2,1,1); plot(t,x)
−4
−3
−2
−1
0
1
2
3
4
5
3
4
5
10
smaller damping factor =−0.3
larger frequency=4.2pi
5
0
−5
y=exp(−0.3*t).*cos(4.2*pi*t);
subplot(2,1,2);plot(t,y);
−10
−15
−5
−4
−3
−2
−1
0
1
2
• On the left-hand-side, σ < 0, est → 0 as t → ∞, and the magnitude |σ| controls the envelope
(damping factor); as |σ| ր increases, the envelope ց decays faster.
• Along the jω-axis, ω controls the angular frequency of the rotating phasor.
I-7
2. Rectangular pulse: ⊓(t/τ )
⊓(t/τ )
1
-
0
− τ2
t
τ
2
3. Triangular pulse: Λ(t/τ )
Λ(t/τ )
1
HH
HH
HH
−τ
τ
0
4. Unit step function: u(t) =
Rt
−∞ δ(λ)dλ
-
=
t
(
1, t ≥ 0
0, t < 0
1
t
0
5. Unit ramp function: r(t) =
Rt
−∞ u(λ)dλ
= tu(t):
t
0
[Ex] tu(t) − 2(t − 1)u(t − 1) + (t − 2)u(t − 2)
1
HH
HH
HH
0
0
-
2
t
I-8
6
Impulse function
6.1
Unit impulse (Kronecker delta function) in discrete time
δ[n] ≡
(
0, n 6= 0
1, n = 0
x
1
x
x
x
-1
0
δ[n]
x
x
x
1
2
3
-
n
1. unit step function:
u[n] ≡
(
0, n < 0
1, n ≥ 0
u[n]
1
x
x
x
x
x
···
x
-1
0
1
2
-
n
3
• difference equation: δ[n] = u[n] − u[n − 1]
• running sum: u[n] =
2. sampling property:
Pn
m=−∞ δ[m]
x[n]δ[n] = x[0]δ[n] similar to differentiation in CT
x[n]δ[n − n0 ] = x[n0 ]δ[n − n0 ] similar to integration in CT
3. sifting property:
x[n] =
X
k
x[k]δ[n − k] · · · · · · convolution of x[n] and δ[n]
= · · · + x[−2]δ[n + 2] + x[−1]δ[n + 1] + x[0]δ[n] + x[1]δ[n − 1] + · · ·
any sequence can be expressed as a sum of scaled/shifted impulses .
6.2
Unit impulse function(Dirac delta function): δ(t), in continuous time
δ(t) = lim
ǫ→0
t
d
1
⊓( ) =
u(t), a pulse with unit area and zero width, located at t = 0
ǫ
ǫ
dt
6
ǫ
- 6
1
ǫ
6
δ(t)
reducing ǫ
area = 1
-
0
-
t
ǫ→0
-
-
0
I-9
t
Properties of δ(t):
R∞
1. sifting:
−∞ x(t) δ(t
− t0 )dt = x(t0 )
6
δ(t − t0 )
x(t0 ) · δ(t − t0 )
6
x(t)
-
Z
x(t) δ(t − t0 )dt =
Z
= x(t0 )
R∞
−2t δ(t
−∞ e
t0
t
x(t0 ) δ(t − t0 )dt
= x(t0 )
Example:
-
≡
t
t0
Z
δ(t − t0 )dt
− 3)dt = e−6
2. Convolution of x(t) with δ(t − t0 ) is x(t − t0 ).
which is useful in proving the sampling theorem and modulation/demodulation.
x(t)
x(t − t0 )
δ(t − t0 )
6
∗
0
Pf. x(t) ∗ δ(t − t0 ) =
R
t0
0
x(τ )δ(t − t0 − τ )dτ =
R
=
0
x(τ )δ(τ − t + t0 )dτ = x(t − t0 ).
Question: What is x(t) ∗ [δ(t − t0 ) + δ(t + t0 )] ∗ [δ(t − t0 ) + δ(t + t0 )]? (demodulation)
3. Other properties:
• x(t)δ(t) = x(0)δ(t).
• x(t)δ(t − t0 ) = x(t0 )δ(t − t0 ).
• δ(t) =
•
•
d
dt u(t).
Rt
−∞ δ(τ )dτ = u(t).
1
δ(at) = |a|
δ(t).
• δ(−t) = δ(t).
• x(t)δ(t − t0 ) = x(t0 )δ(t − t0 ).
•
•
•
R∞
−∞ δ(τ )δ(t
R4
t=0 3δ(t
R4
t=2 δ(t
− τ )dτ = δ(t).
− 2)dt = 3.
− 6)dt = 0.
I - 10
t0
7
Systems
process an input x(t) to produce an output y(t): cause → system → effect
1. Physical models: an electric circuit
2. Block diagrams
• continuous-time: using integrators, adders, and gains (and multipliers)
• discrete-time: using delays, adders, and gains (and multipliers)
3. Mathematical system equations:
• integro-differential equation: y(t) =
dx(t) R t
−∞ [4x(τ )
dt
• difference equation: y[n] = ay[n − 1] + bx[n]
− y(τ )]dτ .
4. Interconnection of systems:
(a) series(cascade):
- S1
- S2
-
(b) parallel:
- S1
6
?- S2
?
-
⊕
-6
(c) feedback:
- S
1
-⊕
6
8
- S
2
-
S3 ?
Classification of Systems
1. Continuous-time or Discrete-time,
2. Analog or Digital,
3. Memoryless: the o/p of a memoryless system at time t0 depends only on its input at the same
time instant t0 . Memory is associated with storage of energy in physical systems and storage
registers in digital computers.
Delay, capacitor, and inductor elements are not allowed for a memoryless system. Thus, a
resistive voltage divider is a memoryless system, while an RC lowpass circuit has memory.
[Ex] y[n] = x2 [n] + x[n], y(t) = 10x(t): memoryless.
2
vi (t) is also memoryless
A voltage divider: vo (t) = R1R+R
2
t
x(τ )dτ, y[n] = x[n − 2], y[n] = (x[n − 1] + x[n] + x[n − 1])/3: with memory.
[Ex] y(t) = −∞
An electric example isR a current source i(t) connected to a capacitor, then the voltage across the
t
i(τ )dτ .
capacitor is v(t) = C1 −∞
R
I - 11
4. Invertible: A system is invertible if the cascade of this system with its inverse system yields an
output which is the input to the first system.
x[n] → system → y[n] → inverse system → w[n] ≡ x[n]
Examples:
(a) System: y(t) = 2x(3t); its inverse system: w(t) = 12 y( 3t )
(b) A running-sum system: y[n] =
tion: w[n] = y[n] − y[n − 1].
Pn
k=−∞ x[k];
its inverse system satisfies a difference equa-
Inverse processing is important in equalization, lossless coding, etc.
5. Causal(nonanticipative) and noncausal: A causal system’s output y(t0 ) can only depends on
the present and past inputs: {x(τ ), τ ≤ t0 }, i.e., the system cannot anticipate the future input.
o/p: y1 (t)
-
i/p: x(t)
0
-
-
causal
o/p: y2 (t)
-
%L
%
L
L
0
t
2
H%
#H
#
Noncausal
-
-2
"
"
e
LL
e
e
-
t
-
t
0
Later, we will show that, for a causal LTI system, its impulse response h(t) = 0, ∀t < 0.
[Ex] y[n] = x[n]− x[n + 1], and y(t) = x(t + 1) are noncausal, because the output y(2) at present
time t = 2 depends on the future input x(3) at t = 3.
• A memoryless system is also causal.
• All physical systems with time as the independent variable are causal. There are practical
systems that are not causal:
– Physical systems for which time is not the independent variable e.g., the independent
variable is (x, y) as in an image. y[n] = (x[n − 1] + x[n] + x[n + 1])/3, take the average
of three neighboring data.
– Processing of signals is not in real time, e.g., the signal has been recorded or generated
in a computer.
6. Stable: A system is stable in the sense of bounded-input, bounded-output(BIBO) if the
output y(t) is bounded for a bounded input x(t), i.e., if |x(t)| ≤ B1 , ∀t, then ∃B2 < ∞, ∋
|y(t)| ≤ B2 ,R∀t. For a stable linear-time-invariant system, its impulse response must be absolutely
∞
|h(t)|dt < ∞. (to be shown later)
integrable: −∞
7. LINEAR: linear combination of inputs lead to linear combination of corresponding outputs
(superposition):
x1 (t) - Linear - y1 (t)
If
and
then
x2 (t) - Linear - y2 (t)
I - 12
ax1 (t) + bx2 (t)
- Linear
- ay1 (t) + by2 (t)
Suppose y1 (t) = H[x1 (t)], and y2 (t) = H[x2 (t)]. If the output due to the input a1 x1 (t) + a2 x2 (t)
is equal to a1 y1 (t) + a2 y2 (t) then the system H is linear, i.e., check if
?
H[a1 x1 (t) + a2 x2 (t)] = a1 y1 (t) + a2 y2 (t)
• Linearity does not allow x2 (t), |x(t)|.
8. TIME-INVARIANT(TI): a delayed input leads to a corresponding delayed output:
If
Time
Invariant
x(t) −→
−→ y(t)
then
x(t − τ ) −→
Time
Invariant
−→ y(t − τ )
for all x(t) and τ .
We need to verify if the following is true:
?
H[x(t − τ )] = {H[x(t)]}|t:t−τ = y(t − τ ), ∀τ
(a) A system is time-invariant if its system parameters are fixed over time.
+
A A A A
h
x(t)
−
R
h
y(t)
C
h
h
d
An RC lowpass filter with constant resistance R and C is time-invariant: RC dt
y(t) + y(t) =
x(t). If the resistance R(t), changes with time, then the system becomes time-varying:
d
y(t) + y(t) = x(t), because its system parameters are not constant over time.
R(t)C dt
(b) Time-invariance does not allow x(2t), x(−t), tx(t).
(c) Ex. A compressor with y[n] = x[M n] is not TI because y[n − n0 ] = x[M (n − n0 )] 6=
x[M n − n0 ]. Be careful in distinguishing x[M (n − n0 )] and x[M n − n0 ].
9
LTI systems
1. Many man-made and naturally occurring systems can be modeled as linear time-invariant (LTI)
systems. Ex. resistors(R), inductors(L), capacitors(C).
2. A fixed-coefficient linear differential equation (made up from RLC components) are LTI. (CT)
3. A fixed-coefficient linear difference equation (delay, gain, adder) are LTI. (Discrete-Time)
4. (Ex) Is the following system linear-time-invariant?
y(t) = x(t)g(t)
where x(t) and y(t) denote the input and output, respectively.
• Linear, yes. Let y1 (t) ≡ x1 (t)g(t) and y2 (t) ≡ x2 (t)g(t). Suppose
x(t) = a1 x1 (t) + a2 x2 (t)
then
H[x(t)] = x(t)g(t)
= a1 x1 (t)g(t) + a2 x2 (t)g(t)
= a1 y1 (t) + a2 y2 (t)
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• Time-varying. Let y(t) = x(t)g(t). Suppose xd (t) = x(t − τ ).
H[xd (t)] = x(t − τ )g(t)
6= y(t − τ ) = x(t − τ )g(t − τ )
• If g(t) = cos ωc t, we call it linear modulation.
9.1
1.
Examples
d
dt y(t)
+ 10y(t) = x(t) is a causal LTI system. y[n] = 0.9y[n − 1] + x[n] is causal LTI, too.
dn−1
dn
dtn y(t) + an−1 dtn−1 y(t) + · · · + a0 y(t)
2. In general,
system.
3. y(t) =
R∞
−∞ x(τ )h(t
m
d
d
= bm dt
m x(t) + · · · + b1 dt x(t) + b0 x(t) is a linear
− τ )dτ is an LTI system. reads as “y(t) is the convolution of x(t) and h(t).”
[Pf] The proof for linearity is omitted.
Let x̂(t) = x(t − λ), then
ŷ(t)
:
=
=
is the response to the input x̂(t)
Z
Zτ
τ
x̂(τ )h(t − τ )dτ
x(τ − λ)h(t − τ )dτ
let s ≡ τ − λ, sorry for so many dummy variables
=
y(t − λ) =
4. (a) y(t) =
(b) y[n] =
(c) y(t) =
y[n] =
(d) y(t) =
R∞
−∞ x(t
P∞
Z
Zs
τ
x(s)h(t − λ − s)ds
x(τ )h(t − λ − τ )dτ, from the definition of y(t) =
−∞ x(τ )h(t
Pn
Rt
− m] =
P∞
m:−∞ x[n
− m]h[m] is LTI. (discrete-time convolution)
− τ )dτ is a causal LTI system.
m:−∞ x[m]h[n
−∞ x(t
x(τ )h(t − τ )dτ
− τ )h(τ )dτ is LTI, too. (commutative law for convolution)
m:−∞ x[m]h[n
Rt
Z
− m] is a causal LTI system.
− τ )h(τ )dτ is a noncausal, linear, time-varying system.
t
0
[Pf] Say h(t) = 1, then y(t) = −∞
x(t − τ )dτ we can see that y(0) = −∞
x(−τ )dτ =
x(u)du,
which
means
that
y(0)
is
the
integral
of
future
input
x(0+)
upto
x(∞).
Non0
causal!
R∞
5. (a)
d
dt y(t)
R
+ ty(t) = x(t) is a linear system.
d
d
y1 (t) + ty1 (t) = x1 (t)
y2 (t) + ty2 (t) = x2 (t)
dt
dt
α1
d
d
y1 (t) + α2 y2 (t) + t[α1 y1 (t) + α2 y2 (t)] = α1 x1 (t) + α2 x2 (t)
dt
dt
d
[α1 y1 (t) + α2 y2 (t)] + t[α1 y1 (t) + α2 y2 (t)] = α1 x1 (t) + α2 x2 (t)
dt
Thus the response to the input α1 x1 (t) + α2 x2 (t) is α1 y1 (t) + α2 y2 (t).
(b)
d
dt y(t)
+ 10y(t) + 5 = x(t) is a nonlinear system.
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R
(c) y(t) = x(t2 ) is linear, noncausal, and time-varying.
y(2) depends on x(4); thus noncausal. Let x̂(t) = x(t − τ ), then ŷ(t) = x̂(t2 ) = x(t2 − τ ) 6=
y(t − τ ) = x((t − τ )2 ); thus time-varying.
(d) y(t) = sin[x(t)] is time-invariant.
(e) y(t) = x(2t) is NOT time-invariant.
[Pf] Suppose x(t) =⇒ y(t) = x(2t). Let x̂(t) = x(t − τ ), then the output from x̂(t) is
ŷ(t) = x̂(2t) = x̂(s)|s=2t = x(s − τ )|s=2t = x(2t − τ )
Compared with the delayed output y(t − τ ):
y(t − τ ) = y(s)|s=t−τ = x(2s)|s=t−τ = x(2(t − τ )) = x(2t − 2τ )
We find that ŷ(t) 6= y(t − τ ), it is NOT TI!
(f) ny[n] + a1 y[n − 1] = x[n] is a linear and time-varying(because of n before y[n]).
(g) y[n] + 0.8y[n − 1] + 5 = x[n] is a nonlinear system(because of the constant 5).
(h) y[n] = x[n2 ] is linear, noncausal, and time-varying.
(i) y[n] = sin(x[n]) is time-invariant.
9.2
Questions
1. If some system: x(t) −→ y(t) is linear, and another system: y(t) −→ z(t) is linear, too, Is the
cascaded system x(t) −→ z(t) linear?
2. If some system: x(t) −→ y(t) is linear, and another system: x(t) −→ z(t) is linear, too, Is the
parallel system x(t) −→ w(t) = y(t) + z(t) linear?
3. Redo the above two questions, when the systems are time-invariant.
4. Is the cascaded connection of two nonlinear systems is nonlinear?
5. (Oppenheim et al.) Consider three systems with the following input/output relationships:
system 1 : y[n] =
(
x[n/2], n : even
0,
n : odd
1
1
system 2 : y[n] = x[n] + x[n − 1] + x[n − 2]
2
4
system 3 : y[n] = x[2n]
Suppose these systems are cascaded in series. Find the input/ouput relationship for the overall
cascaded system. Is it linear? Is it time-invariant?
6. Can you generalize the definitions of linearity and time-invariance for a system processing 3D
signals with 2 independent variables: s(t1 , t2 ), where s = [s1 , s2 , s3 ]?
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