Introducing "Language For Describing Signals and Systems"
Introducing "Language For Describing Signals and Systems"
Introducing "Language For Describing Signals and Systems"
Outline 1.1 ContinuousTime and DiscreteTime Signals 1.2 Elementary Signals 1.3 ContinuousTime and DiscreteTime Systems 1.4 Basic System Properties
1.1
ContinuousTime and DiscreteTime Signals Unied representation of physical phenomena by signals Signal: Function or sequence that represents information. One or more independent variables Continuous or discrete independent variables Examples: time, location, etc.
1.1.1
Mathematical Representation
Continuoustime signals Symbol t for independent variable Use parentheses () Continuoustime signal: x(t) Graphical representation
x(t)
Discretetime signals Symbol n for independent variable Use brackets [] Discretetime signal: x[n] Graphical representation
x[n]
x[0] x[1] x[1]
x[2]
x[2]
4 3 2 1 0 1 2 3
1.1.2
Often classication of signals according to energy and power Terminology energy and power used for any signal x(t), x[n] Need not necessarily have a physical meaning
E(t1, t2) =
t1
|x(t)|2 dt
E(n1, n2) =
n=n1
|x[n]|2
Total energy
E = E(, ) =
|x(t)|2 dt
E = E(, ) =
n=
|x[n]|2
Example: Total energy of the discretetime signal x[n] = with |a| < 1.
an n 0 0 n<0 1 1 |a|2
E =
n=
|x[n]|2 =
n=0
(|a|2)n =
Signal power Consider the timeaveraged signal power Average power of x(t) in interval t1 t t2
t2
P (t1, t2) =
1 t2 t1
|x(t)|2 dt
t1
Analogously P 1 = P (, ) = lim T 2T
T
|x(t)|2 dt
T N
1 = P (, ) = lim N 2N + 1
|x[n]|2
n=N
Classication of signals based on their energy and power Signals with nite total energy E < Zero average power P = 0 Examples: example above, any signal with nite duration Signals with nite average power P < Innite total energy E = if P > 0 Examples: periodic signals, e.g. x(t) = cos(t), x[n] = sin(5n) Signals with innite power P = and innite energy E = Not desirable in engineering applications Examples: x(t) = et, x[n] = n10
1.1.3
Delay: t0, n0 > 0, Advance: t0, n0 < 0 Time reversal Replace t t n n Time scaling Replace t t , IR n n , Z Z x(t) x(t) x[n] x[n] x(t) x(t) x[n] x[n]
Continuoustime case: || < 1 : signal is linearly stretched || > 1 : signal is linearly compressed Time shift, time reversal, and time scaling operations arise naturally in the processing of signals
Example:
x(t)
Signals
x[n]
Time-shifted signals
x(t t0 ) x[n 4]
t0 t n
x(t)
Time-reversed signals
x[n]
x(2/3t)
Time-scaled signals
x[2n]
1.1.4
Periodic Signals
Periodic continuoustime signal x(t) = x(t + T ) , T > 0: Period x(t) periodic with T x(t) also periodic with mT , m IN Smallest period of x(t): Fundamental period T0. Example (T0 = T ):
x(t)
3T 2T
2T
3T
4T
Periodic discretetime signal x[n] = x[n + N ] , Integer N > 0: Period x[n] periodic with N x[n] also periodic with mN , m IN Smallest period of x[n]: Fundamental period N0 . Example (N0 = 4):
x[n]
2 0 1
3 4 5
6 n
10
1.1.5
or
x[n] = x[n]
or
x[n] = x[n]
Necessarily: x(0) = 0 or x[0] = 0 Decomposition of any signal into an even and odd part: x(t) = Ev{x(t)} + Od{x(t)} or x[n] = Ev{x[n]} + Od{x[n]} with 1 1 Ev{x(t)} = (x(t) + x(t)) or Ev{x[n]} = (x[n] + x[n]) 2 2 and 1 1 Od{x(t)} = (x(t) x(t)) or Od{x[n]} = (x[n] x[n]) 2 2
Lampe, Schober: Signals and Communications
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1.2
Elementary Signals Several classes of signals play prominent role model many physical signals serve as building blocks for many other signals serve for system analysis
1.2.1
Complex exponential signal x(t) = Ceat In general, complex numbers C and a (C, a C) Real exponential signal if both a and C real (C, a IR) Periodic complex exponential signal if a = j0 With C = Aej (A, IR): x(t) = Aej(0 t+) Signal is periodic: Aej(0t+) = Aej(0(t+T )+) = Aej(0t+)ej0T where ej0T = 1 Excluding the trivial solution 0 = 0 Fundamental period T0 = 2 |0|
!
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k = 0: 0 (t) constant k = 0: k (t) periodic with fundamental frequency k0 and fundamental period T0/|k| Sets of harmonically related complex exponentials used to represent many other periodic signals General complex exponential signal With C = Aej (A, IR) and a = r + j0 (r, 0 IR) Ceat = Aertej(0t+) = Aertcos(0 t + ) + jAertsin(0 t + ) r > 0: exponentially growing signal r < 0: exponentially decaying signal Sinusoidal signals xc(t) = Acos(0t + ) = Re{Aej(0t+)} and xs(t) = Asin(0 t + ) = Im{Aej(0t+)} xc(t) and xs(t) also have period T0 = 2/|0|, of course.
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Periodic signals have innite total energy but nite average power. Exponential x(t) = ej0t Energy over one period T0
T0
Eperiod =
0
|ej0t|2 dt = T0
Eperiod =1 T0
|ej0t |2 dt = 1
T
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1.2.2
Complex exponential signal x[n] = Cn (= Cen , = e ) Real exponential signal if both C and real General complex exponential signal With C = Aej and = ||ej0 (A, , 0 IR) x[n] = A||n ej(0n+) = A||n cos(0n + ) + jA||nsin(0n + ) || > 1: exponentially growing signal || < 1: exponentially decaying signal || = 1: x[n] = Aej(0n+) = Acos(0n + ) + jAsin(0n + ) Sinusoidal signal xc[n] = Acos(0 n + ) = Re{Aej(0n+)} and xs[n] = Asin(0n + ) = Im{Aej(0n+)} Both Aej(0n+) and Acos(0n + ) are discretetime signals with nite average power but innite total energy.
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Certain properties Important dierences between discretetime and continuoustime complex exponentials 1. Increase frequency 0 by integer multiples of 2 ej(0+m2)n = ej0n ejm2n = ej0n Observation: same exponential for frequency 0 and frequencies 0 2, 0 4, . . . Conclusion: sucient to consider frequency interval of length 2 Usually: intervals 0 0 < 2 and 0 < used Example: Fig. 1.27 in text book 2. Periodicity: period N > 0 or ej0N = 1 0 N = 2m, m 0 = 2 N
!
where m is integer Observation: ej0n is periodic if 0/(2) is a rational number, and is aperiodic otherwise. 3. Fundamental period N0: N0 = m 2 0
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Example: x(t) = cos(8t/31) 0 = Fundamental period T0 = 2/0 = x[n] = cos(8n/31) (= x(t = n)) 0 = Periodic? Fundamental period N0 = m(2/0) = for m = x[n] = cos(n/6) 0 = Periodic? Fundamental period N0 = m(2/0) = for m =
Set of harmonically related discretetime periodic exponentials k [n] = ejk(2/N)n, Common period N Observation: k+N [n] = ej(k+N)(2/N)n = ejk(2/N)nej2n = k [n] Only N distinct complex exponentials 0[n], 1[n], . . ., N1 [n]. k = 0, 1, . . .
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1.2.3
Unit impulse sequence (or unit impulse or unit sample) [n] = 1, 0, n=0 n=0
u[n]
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Relation between [n] and u[n] First order dierence [n] = u[n] u[n 1] Running sum
n
u[n] =
m=
[m]
Sampling property of unit impulse x[n][n n0] = x[n0][n n0] 1.2.4 The ContinuousTime Unit Impulse and Unit Step Functions
t>0 t<0
Note: discontinuity at t = 0 Unit impulse function (unit impulse, Dirac delta impulse) (t) = ?, 0, t=0 t=0
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Remark: We use the short-hand notation: dx(t) = x(t) dt Relation between (t) and u(t) First order derivative (t) = u(t) Running integral
t
u(t) =
( ) d
Formal diculty: u(t) is not dierentiable in the conventional sense because of its discontinuity at t = 0. Some more thoughts on (t) Consider functions u (t) and (t) instead of u(t) and (t):
u (t) 1 t
1
(t)
where
(t) = u(t)
t
u (t) =
( ) d
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(t) :
3 (t)
1 3
2 (t)
1 2 1 1
1 (t)
3 2
Observe: Area under (t) always 1 (t) is an innitesimally narrow impulse with area 1. (t) = lim (t)
0
( ) d = 1
Representation
(t) (t t0 ) a 1 1 a(t)
t0
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x( )( t0) d = x(t0)
Linearity
(a( ) + b( ))x( ) d =
a( )x( ) d + b( )x( ) d
(a )x( ) d =
(at) =
22
( )x( ) d = (t)x(t)
( )x( ) d = x(0)
( )x( ) d = x(0)
t(t) = (t) Remark: More formal discussion of the unit impulse (t) in text books on generalized functions or distributions.
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1.3
System: Entity that transforms input signals into new output signals One or more input and output signals Continuoustime system transforms continuoustime signals Discretetime system transforms discretetime signals Formal representation of inputoutput relation Continuoustime system x(t) y(t) Discretetime system x[n] y[n] Remark: Another popular notation that you may nd in books is y(t) = S{x(t)}, where S{} represents the system operator. Pictorial representation of systems
x(t)
Continuoustime system
y(t)
x[n]
Discretetime system
y[n]
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1.3.1
Quadratic system y(t) = (x(t))2 System represented by a rst order dierential equation y(t) + ay(t) = bx(t) with constants a and b Delay system y[n] = x[n 1] System described by a rst order dierence equation y[n] = ay[n 1] + bx[n] with constants a and b 1.3.2 Interconnections of Systems
Often convenient: break down a complex system into smaller subsystems Series (cascade) interconnection
Input System 1 System 2 Output
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Parallel interconnection
System 1 Input System 2 Output
Example: Diversity transmission: transmission of the same signal over two antennas and receiving it with one antenna Feedback interconnection
Input System 1 Output
System 2
Examples: Closed-loop frequency/phase/timing synchronization in communications, human motion control 1.4 Basic System Properties Simple mathematical formulation of basic (physical) system properties Classication of systems For conciseness: only denitions for continuous-time systems Replacing (t) by [n] denitions for discrete-time systems
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1.4.1
Linearity
Let x1(t) y1(t) and x2(t) y2(t) Linear system if 1. Additivity x1(t) + x2(t) y1 (t) + y2 (t) 2. Homogeneity ax1(t) ay1(t) , a C
ak yk (t)
Not linear systems are referred to as nonlinear. Example: 1. System y(t) = tx(t) is linear. To see this let x1(t) y1 (t) = tx1(t) x2(t) y2 (t) = tx2(t) and x3(t) = ax1(t) + bx2(t) ,
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and check y3(t) = tx3(t) = tax1(t) + tbx2(t) = ay1 (t) + by2(t) , i.e., ax1(t) + bx2(t) ay1 (t) + by2(t) 2. System y[n] = (x[n])2 is nonlinear. To see this let x1[n] y1[n] = (x1[n])2 x2[n] y2[n] = (x2[n])2 and check additivity for input x3[n] = x1[n] + x2[n] y3[n] = (x3[n])2 = y1[n] + y2 [n] + 2x1[n]x2[n] = y1[n] + y2[n] 3. System y(t) = (x(t)) is ?
1.4.2
Time Invariance
Time invariant system if behavior and characteristics are time-invariant, i.e., identical response to same input signal no matter when input signal is applied x(t t0) y(t t0) Example: 1. The system y(t) = (x(t))2 is ? 2. The system y[n] = nx[n] is ?
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Remark: Linear and time-invariant (linear time-invariant (LTI)) systems play a prominent role for system modeling and analysis. The importance of complex exponentials derives from the fact that they are eigenfunctions of LTI systems. 1.4.3 Systems with and without Memory
Memoryless system if output signal depends only on present value of input signal Otherwise, a system is said to possess memory or to be dispersive. Example: Memoryless systems x[n] , A x[n] A 1. Limiter: y[n] = A , x[n] < A A , x[n] > A 2. Amplier: y(t) = A x(t) Systems with memory
n
1. Accumulator: y[n] =
k=
x( ) d
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1.4.4
Invertible system if bijective transformation x(t) y(t) from input to output In this case an inverse system y(t) w(t) = x(t) exists
y(t) x(t)
System
Inverse system
w(t) = x(t)
Example: Invertible systems 1. Amplier: y(t) = Ax(t), A = 0 1 Inverse system: w(t) = A y(t) (=Amplier) 2. Accumulator: y[n] = y[n 1] + x[n] Inverse system: w[n] = y[n] y[n 1] (=Dierentiator) Noninvertible systems x[n] , A x[n] A 1. Limiter: y[n] = A , x[n] < A A , x[n] > A 2. Slicer: y[n] = 1 , x[n] 0 1 , else
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1.4.5
Causality
Causal system if output at any time depends only on past and present values of the input If x1(t) = x2(t) for t t0 then y1(t) = y2 (t) for t t0 , Implication: Causal+Linear: If x1(t) = 0 for t t0 then y1(t) = 0 for t t0, t0 Example: Causal system
n
t0
Accumulator: y[n] =
k=
x[n k]
k=N
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1.4.6
Stability
Consider boundedinput boundedoutput (BIBO) stability Stable system if for any bounded input signal |x(t)| Bx < , the output signal is bounded |y(t)| By < , Example: Stable system 1 Averager: y[n] = 2N + 1
N
x[n k]
k=N
Bounded input |x[n]| < Bx bounded output |y[n]| < By = Bx Instable system
t
Integrator: y(t) =
x( ) d
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Example: The rst Tacoma Narrows suspension bridge collapsed due to wind-induced vibrations, November 1940.