Summer 2008
Signals & Systems
S.F. Hsieh
I. Introduction to Signals and Systems 1
Goal Goal of Sign Signal alss & Syst System emss
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 ( f (t) speech(voltage/current picked up from microphone, a computer file xxx .wav, stock market market index, index, ECG(ElectroC ECG(ElectroCardioG ardioGram); ram); f ( f (x, y )/f ( /f (x, y ) image xxx .gif or xxx.jpg; f ( f (t,x,y) t,x,y ) video xxx.mpg.
• Systems(mapping) proce process ss input input signal signalss
to produc producee outpu outputt signa signals ls.. They They can be hardhardwares(DSP processors/circuits/automobile, economic system) or softwares(programs).
2
Clas Classi sific ficat atio ions ns of sign signal alss 1. Digital/ana Digital/analog: log: the amplitude is discrete(qu discrete(quanti antized) zed) or not. 2. Continuous-/Discrete-time: Continuous-/Discrete-time: (MATLAB (MATLAB commands: plot, plot, stem)
• Continuous-time signal: x(t), −∞ < t < ∞, where the time-v time-variable t is continuous. nT ), n = 0, ±1, ±2, . . ., ., where T represents T represents the sampling • Discrete-time signal:1 x[n] ≡ x(nT ), time. Note that x that x[3 [3 2 ] 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
−∞
|x(t)|2dt
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 Total energy over the time interval t 1 t t2 is tt x(t) 2 dt. Discrete-time: total energy over the time interval n1 n n2 is nn=n x[n] 2 . – average power: 1 T/2 P lim x(t) 2 dt T →∞ T −T /2 If x(t) is periodic, then its average power becomes 1 T/2 P = x(t) 2 dt T −T /2
∗
|
≤ ≤
∗
≡
∗
2
1
| ≤ ≤
2
1
| |
| |
| |
∗ The rms(root-mean-square) value of x(t) is √ P .
• Definitions:
– x(t) is an energy signal if it has a finite energy E x < and its power P x = 0. – x(t) is a power signal if it has a finite power P x < , and energy E x = .
∞
• (Lathi Ex 1.2, p 72)
∞
∞
2
– Power of a single real sinusoid x(t) = C cos(ω0 t + θ) is P x = C 2 , where C is a positive amplitude. – Power of a single exponential y(t) = De jω t is P y = D 2 , where D is a complex number comprised of amplitude and phase. – Power of sum of orthogonal complex exponentials z(t) = i Di e jω i t with distinct frequencies ω i’s is equal to the sum of individual powers P z = i P i = 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., P z = P x + P y ? 0
| |
• Examples: (a) (b) (c) (d) (e)
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x(t) = e −3t u(t) is an energy signal with energy 1/6. x(t) = Ae j2παt is a power signal with power A2 . A rectangular pulse A (t/τ ) of amplitude A and duration τ has energy A 2 τ . A (t/τ ) cos(ω0 t + θ) has energy A 2 τ /2. t−nT 2 A rectangular pulse train ∞ n:−∞ ( τ ), T 0 > τ , is a power signal with P = A τ /T 0
⊓
·
⊓
⊓
I-2
0
(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) = x 1 (t) + x2 (t) ✘
✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✟ ✟ ✟ x1 (t) ✟ ❳ ❳ ❳ ✟ ✟ ✟ ❳ ❳ ❳ x2 (t) ✟ ❳ ❳ ✟ ❳ ✲
t
[Ex] Given x(t), plot y(t) = 2x(t) + 3. (“multiply/divide first, then add/subtract”)
−
x(t)
1
✓ ✓ ✓ -1
0
-1
0
❆ ❆ ❆ -2 ❆
✲ t 1
3
1
✲
t
❅ ❅ 1
−2x(t)
✲ -1
2. change of Time-variable: Assume that x(t) looks like x(t)
1
✁ ✁ ✁ ✁ ✁
-1
✲ 1
t
I-3
0
1
t
(a) time-scale: x(t) =
⇒ x(αt), where α > 0
• α > 1, time-compression: 1
✄ ✄ ✄ ✄ ✄
− 12
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 12 ) = x(1), and y( 12 ) = x( 2 12 ) = x( 1).
−
−
−
(b) time-reversal(inversion): x(t) = 1
❆ ❆ ❆ ❆ ❆
⇒ x(−t) x(−t) ✲
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) ✁ ✁ ✁ ✁ ✁
x(t + β )
✲
−β
0
• β < 0, right-shifted(delayed): ✁ ✁ ✁ ✁ ✁
x(t
− β )
✲
β
0
t
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.
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−
−
−
(d) Combined operation: x(t) = [method 1]:
⇒ x(αt + β ) ,
• 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 + x(α[t + β ]) = x(αt + αβ ) = z(t)! Note:
β α ])
= x(αt + β ) = z(t), while w′ (t) = v(t + β ) =
• if you have difficulty in combining w(t) and v(t), you can use more dummy variables to avoid confusion:
v(s) = x(αs), w(t) = v(t + so that
β ) α
β ]) = x(αt + β ) α α [Method 1] is preferred due to its algebraic simplicity. w(t) = v(s) s=t+ β = x(α[t +
•
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[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.)
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−
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)
x(t)
1
✁ ✁ ✁ ✁ ✁
✲
-1
1
• Right shift by 3, x(t − 3)
x(t
1
✁ ✁ ✁ ✁ ✁
2
3
t
− 3) ✲
4
t
• Compress by 2, x(2t − 3) ✄ ✄ ✄ ✄ ✄ 1
✲ 2
t
• (Check) let x(2t let x(2t let x(2t
− 3) = x(0), − 3) = x(−1), − 3) = x(2),
we have t = 3/2 we have t = 1 we have t = 5/2
Similarly,
• Given x(t)
x(t)
1
✁ ✁ ✁ ✁ ✁
-1
✲ 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) ❊ ❊ ❊ ❊ ❊ 1
✲
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, ★❝ ★ ❝ ★ ❝
• Odd function(signal): x(t) = −x(−t), ∀ t, anti-symmetric about the origin, ✻ ❜ ❜ ❜ ❜ ❜
◗ ◗ ◗ ◗ ◗
• (Fact) Every signal x(t) can be decomposed as a sum of even and odd signals: x(t) = xe (t) + xo (t), where x e (t) ≡ E v {x(t} ≡ x(t)+x(−t) and x o (t) ≡ O d{x(t} ≡ x(t)−x(−t) . 2 2 2
❆ x(t) ❆ ❆ ❆
✻ x (t) ❧ e ❧ ❧
1 =
2. Periodic signals: x(t) = x(t + nT ),
+
∀t. T is its period.
✻ x (t) ❅ o ❅
❅ ❅
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
≡ Ae jφ e(σ+ jω)t ······ s ≡ σ + jω in Laplace transform Cz n ≡ Ae jφ (re jω )n ······ z ≡ re jω in Z transform
x[n] =
(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 = 0,
i. σ = 0 or r = 1: x(t) = Ae jφ e jωt = A cos(ωt + φ) + jA sin(ωt + φ) or x[n] = Ae jφ e jωn is a non-damping cisoid (complex sinusoid). will see in Fourier transforms. σt j(ωt+φ) ii. σ > 0 or r > 1: x(t) = Ae e is an exponentially increasing(unstable) cisoid σt j(ωt+φ) iii. σ < 0 or r < 1: x(t) = Ae e is an exponentially damped(decreasing) cisoid
······
[Euler’s formula] e jωt
cos ωt + j sin ωt and e− jωt cos ωt j sin ωt 1 jωt 1 cos ωt = (e + e− jωt ) and sin ωt = (e jωt e− jωt ) 2 2 j
≡
≡
−
−
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 j ω-axis, ω controls the angular frequency of the rotating phasor. I-7
2. Rectangular pulse:
⊓(t/τ ) ⊓(t/τ )
1
✲ 0
− τ 2
t
τ 2
3. Triangular pulse: Λ(t/τ ) Λ(t/τ ) 1
✟❍ ❍ ✟ ✟ ❍ ❍ ✟ ✟ ❍ ✟ ❍
−τ
0
4. Unit step function: u(t) =
t −∞ δ (λ)dλ =
✲
τ
t 1, t 0 0, t < 0
≥
1
t
0 5. Unit ramp function: r(t) =
t −∞ u(λ)dλ = tu(t):
✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ t
0 [Ex] tu(t)
− 2(t − 1)u(t − 1) + (t − 2)u(t − 2) 1
✟❍ ❍ ✟ ✟ ❍ ❍ ✟ ✟ ❍ ✟ ❍ 0
0
✲
2
t
I-8
6
Impulse function
6.1
Unit impulse (Kronecker delta function) in discrete time δ [n]
≡
0, n = 0 1, n = 0
1 ①
①
①
①
-1
δ [n] ①
0
1
①
✲
①
2
n
3
1. unit step function: u[n]
≡
0, n < 0 1, n 0
≥
u[n] 1 ①
①
①
①
①
···
①
-1
0
1
2
✲
n
3
• difference equation: δ [n] = u[n] − u[n − 1] • running sum: u[n] = nm=−∞ δ [m]
2. sampling property:
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]δ [n
− k] ······ convolution of x[n] and δ [n] k ··· + 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 1 ǫ→0 ǫ
δ (t) = lim
⊓ ( ǫt )
=
d u(t), a pulse with unit area and zero width, located at t = 0 dt
✻ǫ ✲ ✛
✻ 1 ǫ
✻ δ (t)
reducing ǫ
area = 1 0
ǫ
✲ t
✲
→0
✲ ✲
✲ 0
I-9
t
Properties of δ (t): ∞ −∞ x(t) δ (t
1. sifting:
δ (t
− t0)dt = x(t0) ✻
− t0 )
x(t0 ) δ (t
· − t0 )
✻
x(t)
✲ t0
x(t) δ (t
− t0)dt
= =
✲
≡
t
t0
t
x(t0 ) δ (t
− t0 )dt x(t0 ) δ (t − t0 )dt
= x(t0 )
2.
∞ −2t δ (t −∞ e
− 3)dt = e−6 Convolution of x(t) with δ (t − t0 ) is x(t − t0 ).
Example:
which is useful in proving the sampling theorem and modulation/demodulation. x(t)
δ (t
✻ 0
∗
t0
0
x(t
− t0 ) = 0
∗ − t0) = x(τ )δ (t − t0 − τ )dτ = 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) Pf. x(t) δ (t
3. Other properties:
• x(t)δ (t) = x(0)δ (t). • x(t)δ (t − t0) = x(t0)δ (t − t0). • δ (t) = dtd u(t). t • −∞ δ (τ )dτ = u(t). • δ (at) = |a|1 δ (t). • δ (−t) = δ (t). (t − t0 ) = x(t0 )δ (t − t0 ). • x(t)δ ∞ • −∞ δ (τ )δ (t − τ )dτ = δ (t). 4 3δ (t − 2)dt = 3. • t=0 4 δ (t − 6)dt = 0. • t=2
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− t0 ) 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:
t • integro-differential equation: y(t) = dx(t) −∞ [4x(τ ) − y(τ )]dτ . dt • difference equation: y[n] = ay[n − 1] + bx[n] 4. Interconnection of systems: (a) series(cascade):
✲ S 1
✲ S 2
✲
(b) parallel:
✲ S 1 ✻ ✲ ❄✲ S 2
✲ ❄ ✲ ✲ ✻
⊕
(c) feedback:
✲ S 1
✲
⊕
✻ ✛
8
✲ S 2
✲
S 3 ✛ ❄
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] = x 2 [n] + x[n], y(t) = 10x(t): memoryless. A voltage divider: vo (t) = R R+R vi (t) is also memoryless 2
1
t −∞ x(τ )dτ,y[n]
2
[Ex] y (t) = = x[n 2], y[n] = (x[n 1] + x[n] + x[n 1])/3: with memory. An electric example is a current source i(t) connected to a capacitor, then the voltage across the 1 t capacitor is v(t) = C −∞ i(τ )dτ .
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−
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−
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) = 21 y( 3t ) (b) A running-sum system: y[n] = tion: w[n] = y[n] y[n 1].
−
−
n 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
✲
✪ ▲ ✪ ❍ ★ ❍✪ ▲ ★ ▲ ✲ ▲ ✔ ▲ ✔ 0
✲
causal
✧ ✧
o/p: y2 (t)
t
✲
2
✲
Noncausal
❡ ❡ ❡
✲
0
-2
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 , t. For a stable linear-time-invariant system, its impulse response must be absolutely ∞ integrable: −∞ h(t) dt < . (to be shown later)
| |≤
∀
| |≤
| |
∀
∃
∞ ∋
∞
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)
t
t
Suppose y 1 (t) = [x1 (t)], and y 2 (t) = [x2 (t)]. If the output due to the input a 1 x1 (t) + a2 x2 (t) is equal to a1 y1 (t) + a2 y2 (t) then the system is linear, i.e., check if
H
H
H
? H[a1x1(t) + a2x2(t)] = a 1 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 x(t)
Time Invariant
−→
−→ 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.
❆ ✁❆ ✁ ❆✁ ❆✁
+ ❤ x(t)
−
R
❤
y(t)
C
❤
❤
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 R(t)C dt y(t) + y(t) = x(t), because its system parameters are not constant over time. (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 n 0 ] = x[M (n n 0 )] = x[M n n0 ]. Be careful in distinguishing x[M (n n0 )] and x[M n n0 ].
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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 y 1(t) ≡ x1(t)g(t) and y 2(t) ≡ x2(t)g(t). Suppose x(t) = a 1 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) I - 13
• Time-varying. Let y (t) = x(t)g(t). Suppose x d(t) = x(t − τ ). H[xd(t)] = x(t − τ )g(t) = y(t − τ ) = x(t − τ )g(t − τ ) • If g(t) = cos ωct, we call it linear modulation . 9.1 1.
Examples d dt y(t) + 10y(t)
dn dn−1 dtn y(t) + an−1 dtn−1 y(t) +
2. In general, system. 3. y(t) =
= x(t) is a causal LTI system. y[n] = 0.9y[n
∞ −∞ x(τ )h(t
m
··· + a0y(t) = bm dtd
m
− τ )dτ is an LTI system.
− 1] + x[n] is causal LTI, too. d x(t) + ··· + b1 dt x(t) + b0 x(t) is a linear
reads as “y(t) is the convolution of x(t) and h(t).”
[Pf] The proof for linearity is omitted. Let x ˆ(t) = x(t yˆ(t)
− λ), then : = =
= y(t
− λ)
(c) y(t) y[n] (d) y(t)
− τ )dτ x(τ − λ)h(t − τ )dτ τ let s ≡ τ − λ, sorry for so many dummy variables x(s)h(t − λ − s)ds s x(τ )h(t − λ − τ )dτ, from the definition of y (t) = x(τ )h(t − τ )dτ τ τ
x ˆ(τ )h(t
∞ τ )h(τ )dτ is LTI, too. (commutative law for convolution) −∞ x(t = ∞ m] = ∞ m]h[m] is LTI. (discrete-time convolution) m:−∞ x[m]h[n m:−∞ x[n t = −∞ x(τ )h(t τ )dτ is a causal LTI system. = nm:−∞ x[m]h[n m] is a causal LTI system. t = −∞ x(t τ )h(τ )dτ is a noncausal, linear, time-varying system. t 0 Say h(t) = 1, then y(t) = −∞ x(t τ )dτ we can see that y(0) = −∞ x( τ )dτ
4. (a) y(t) = (b) y[n]
=
is the response to the input x ˆ (t)
−
−
−
−
−
−
[Pf] = ∞ 0 x(u)du, which means that y(0) is the integral of future input x(0+) upto x( ). Noncausal!
5. (a)
d dt y(t) +
−
ty(t) = x(t) is a linear system.
d d y1 (t) + ty1 (t) = x1 (t) y2 (t) + ty2 (t) = x 2 (t) dt dt α1
d d y1 (t) + α2 y2 (t) + t[α1 y1 (t) + α2 y2 (t)] = α 1 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 α 1 x1 (t) + α2 x2 (t) is α 1 y1 (t) + α2 y2 (t). (b)
d dt y(t) + 10y(t)
+ 5 = x(t) is a nonlinear system.
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− ∞
(c) y(t) = x(t2 ) is linear, noncausal, and time-varying. y(2) depends on x(4); thus noncausal. Let x ˆ (t) = x(t 2 y(t τ ) = x((t τ ) ); thus time-varying.
−
−
− τ ), then yˆ(t) = xˆ(t2) = x(t2 − τ ) =
(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 x ˆ (t) = x(t
− τ ), then the output from xˆ(t) is yˆ(t) = x ˆ(2t) = x ˆ(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 yˆ(t) = 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) cascaded system x(t)
−→ y (t) is linear, and another system: −→ z(t) linear? 2. If some system: x(t) −→ y (t) is linear, and another system: parallel system x(t) −→ w(t) = y(t) + z(t) linear?
y(t)
−→ z (t) is linear, too, Is the
x(t)
−→ z (t) is linear, too, Is the
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 system 2 : y[n] = x[n] + x[n 2 system 3 : y[n] = x[2n]
− 1] + 14 x[n − 2]
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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