Signals & Systems Notes PDF

Signals & Systems 503371-3 Chapter1: Introduction to Signals and Systems Based on the Book: Sabah, N.H. (2008). Electric Circuits and Signals (1st ed.). CRC Press. 1 Outline of Chapter 1 2        Introduction Continuous-time and discrete-time signals Transformations of the independent variabl...

Signals & Systems 503371-3 Chapter1: Introduction to Signals and Systems Based on the Book: Sabah, N.H. (2008). Electric Circuits and Signals (1st ed.). CRC Press. 1 Outline of Chapter 1 2        Introduction Continuous-time and discrete-time signals Transformations of the independent variable Exponential and sinusoidal signal The unit impulse & unit step functions Continuous-time and discrete-time systems Basic system properties What is a signal? 3    The concept of signal refers to the time, space or other types of variations in the physical state of an object, phenomenon, entity etc. The quantification of this state is used to represent, store or transmit a message. Signal is a function of one or more variables that conveys information about a physical phenomenon. Examples of signal include:     Electrical signals: Voltages and currents in a circuit Acoustic signals: Acoustic pressure (sound) over time Mechanical signals: Velocity, acceleration of a car over time, Video signals: Intensity level of a pixel (camera, video) over time What is a System? 4     A system is an entity that manipulates one or more signals that accomplish a function, thereby yielding new signals. Systems process input signals to produce output signals. The input/output relationship of a system is modeled using mathematical equations. Examples of a system :      Electric circuits (Input: Voltage, Output: Current) Mass spring system (Input: Force, Output: displacement) Automatic speaker recognition (Input: Speech, Output: Identity) Electronics, radio, TV, radar, etc. Communication systems, filter, equalizer, synthesizer, etc. Continuous & Discrete-Time Signals 5  Continuous-time (CT) signals   Most signals in the real world are continuous time (e.g., voltage, velocity). CT signal is denoted by x(t), where the time interval may be bounded (finite) or infinite x(t) t Discrete-time (DT) signals  Some real world and many digital signals are discrete time, as they are sampled. E.g. pixels, daily stock price (anything that a digital computer processes). DT signal is denoted by x[n], where n is an integer value that varies discretely x[n] x [n ]  x (nT ), T is sampling period n Continuous & Discrete-Time Signals 6   Example of continuous signal : voltage, speech, ECG,… Example of discrete signal (T=0.4s) e.g. pixels, daily stock price (anything that a digital computer processes) Signal Energy & Power 7      It is often useful to characterise signals by measures such as energy and power. For example, the instantaneous power of a resistor is: 1 2 p(t )  v(t )i(t )  v (t ) R The total energy expanded over the interval [t1, t2] is: t2 t2 1 E   p (t )dt   v 2 (t )dt joules t1 t1 R The average power is: t2 t2 1 1 1 2 ( ) (t )dt watts P p t dt  v   t t t 2  t1 1 t 2  t1 1 R How are these concepts can be defined for any continuous or discrete time signal? Signal Energy and Power 8  Normalized total energy of a CT signal x(t) over [t1, t2] is: t2 2 E   x(t ) dt t1   where |.| denote the magnitude of the (possibly complex) number. Normalized total energy of a DT signal x[n] over [n1, n2]: E  nn x[n] n2 2 1  By dividing the quantities by (t2-t1) and (n2-n1+1), respectively, gives the normalized average power, P t2 1 2 ( ) P x t dt  t2  t1 t1 Energy and Power over Infinite Time 9  For many signals, we’re interested in examining the power and energy over an infinite time interval (-∞, ∞). The total energy in CT& DT are: E  limT   T 2 x(t ) dt   T   2 x(t ) dt E  lim N   n  N x[n]   n  x[n] N   2  2 If the integrals or sums do not converge (e.g., x(t) or x[n] = nonzero constant value for all time), then the energy of such signal is infinite. The averaged power over an infinite interval in CT & DT are: P  limT  P  lim N  1 2T  T T 2 x (t ) dt N 1 2 x [ n ]  2N  1 n  N Energy and Power over Infinite Time 10  Two important sub-classes of signals:  Finite total energy, E∞ < ∞. Therefore the signal have zero average power. E P  limT    0, 0  E   and P  0 2T 0  t 1 1 If signal x (t )   others 0 E   1 & P  0  Finite average power, when P∞ > 0, then the signal have infinite total energy (E∞ = ∞) because of integration or summing over an infinite time interval. 0  P   and E   Energy and Power 11     An energy signal has zero power A power signal has infinite energy. Periodic signals and random signals are usually power signals (infinite energy & finite average power). Signals that are both deterministic and aperiodic (bounded finite duration) are usually energy signals. Deterministic and Random Signals 12   A signal is deterministic if we can define its value at each time point as a mathematical function (e.g., sine wave) A signal is random if it cannot be described by a mathematical function (can only define statistics) (e.g., electrical noise generated in an amplifier of a radio/TV receiver). Example of Energy Signals 13 x(t) x[n] 5 5 0 0 -5 -5 0 5 -5 -5 x(t) 5 0 0 0 5 x[n] 5 -5 -5 0 5 -5 -5 0 5 Examples of Power Signals 14 1 3 0.8 2 0.6 0.4 amplitude amplitude 1 0 -1 0.2 0 -0.2 -0.4 -0.6 -2 -0.8 -3 0 1 2 3 4 5 6 time (s) Gaussian noise 7 -3 x 10 -1 0 1 2 3 4 5 6 time (s) Sinusoidal signals 7 -3 x 10 T Ex. x (t )  A sin(0t   ) is periodic 2 with period T  0 1 sin 2 ( x )  1  cos  2x 2  1 P   A2 sin 2 (0t   )dt T 0 A20  2  2 / 0 0 0  1 1  2  2 cos(20t  2 )  dt A2  Power Signal 2 Energy and power Signals Ex. 15  Ex. x (t )  A e  0  t t 0 t 0 E   A 2 e 2t dt A 2t e  2  0 A t 0 2 x (t ) A2 A2 (1))   0  ( 2 2  L 2 2t  E  lim   A e dt  L  0  A 2 2t L A 2 2 L A2 e 0   lim e  1   lim   L  L  2 2 2 L A 2 2 L 1  2 2t  e  1  0 P  lim A e dt   lim   L  2L L  4L 0  Energy and power Signals Ex. 16  Ex. A et t 0 x (t )   A u (t ) t  0 L 0 2  2 2t E  lim   A dt   A e dt  L  0 L  1    lim A 2  L  1  e 2 L     L  2   0 L  1  2 2 2t P  lim   A dt   A e dt  L  2L 0 L  A2  1 A2 2 L   lim L  1  e    L  2L  2   2 x (t ) A t Energy and power Signals Ex. 17  Ex. E   A  2 e 2a t 2 A 2a t dt   e 2a  x (t )  A e a t A  A2 A2  0  1  2a 2a L 2  1  A 2 a t  2 P  lim A e dt   lim   L  2L L  2L L  0 2 A e2aL  e2aL   0  lim L  4aL 0  1 2a t   2a e L L   t Energy and power Signals Ex. 18  Ex. E  /2  2 x (t )dt   /2  /2   /2 2 2  /2 A A dt  A t  /2  A2   2 x (t ) 0 1  L 2 , ( ) P  lim x t dt  L  2 L    L 1  /2 2  lim A dt ,  L  2 L  /2 A2t  /2 A2 A2 | /2  lim  lim   0. L  2 L L  2 L  Since E = A2/ τ and P = 0. Therefore, the signal is an energy signal.  2 t Time Shift Signal Transformations 19   A central concept in signal analysis is the transformation of one signal into another signal. Of particular interest are simple transformations that involve a transformation of the time axis only. A linear time shift signal transformation is given by: y(t )  x(at  b)  Given x(at + b), depending on the values of a and b we have:    Time scaling (stretch & compression) Time reversal Time shift Time Shift Signal Transformations 20  We will investigate x(at + b) given x(t) for different values of a and b:   at  b t    b a where b represents a signal offset from 0, and the a parameter represents the signal scaling (stretch & compression) & reversal       if 0 < a < 1: then linearly stretched signal Time scaling if a > 1: then linearly compressed signal if a < 0: then reversed in time. Time reversal if b > 0: then time advance (the signal shifts left) Time shift if b < 0: then time delay (the signal shifts right) Note: It is important to shift first and then compress/stretch. Time Shift Signal Transformations 21 Operation of reflection: (a) continuous-time signal x(t) and (b) reflected version of x(t) about the y-axis. Time Shift Signal Transformations 22 Time-scaling operation; (a) continuous-time signal x(t), (b) compressed version of x(t), and (c) stretched version of x(t) Time Shift Signal Transformations 23 Time-shifting operation: (a) continuous-time signal in the form of a rectangular pulse of amplitude, symmetric about the y-axis; and (b) time-shifted version of x(t). Time Shift Signal Transformations 24 Time-shifting & scaling operations: (a) Rectangular pulse x(t), symmetric about the y-axis. (b) v(t) is a time-shifted version of x(t). (c) y(t) is a compression version of v(t). Time Signal Properties 25   Periodic signals: a signal is periodic if it repeats itself after a fixed period T, otherwise the signal is called non-periodic or aperiodic. Even and odd signals: A signal is even if it can be reflected around y-axis : x(-t) = x(t) (e.g., cos(t) is even signal).  A signal is odd if symmetric around origin t = 0: x(-t) = -x(t) (e.g., sin(t) is odd signal). and sinusoidal signals: a signal is (real) exponential Exponential    if it can be represented as x(t) = Ceat. A signal is (complex) exponential if it can be represented in the same form, where C and a are complex numbers. Unit impulse & unit step functions: A pulse signal is one which is nearly completely zero, apart from a short spike, δ(t). A unit step signal u(t) is zero up to a certain time and then it has a constant value after that time. Periodic Signals-CT 26  Continuous-time signal x(t) is periodic if it has the property that there is a positive value of T for which x(t )  x(t  T )  for T  0 for all t. If x(t) is periodic with period T, then x(t )  x(t  mT ) for all t & for all integers m.    Thus, x(t) is also periodic with period 2T, 3T,...... The fundamental period T0 of x(t) is the smallest positive value of T. A constant signal x(t) = 5 is periodic with any real period, the fundamental period T0 is undefined. Examples: cos(t + 2) = cos(t) and sin(t + 2) = sin(t) are both periodic with period 2. Periodic Signals-DT 27  Discrete-time signal x[n] is periodic with period N if it is unchanged by a time shift of N, where N is an integer. x[n]  x[n  N ]  for N  0 for all n. If x[t] is periodic with period N, then x[n]  x[n  mN ] for all n & for all integers m.   Thus, x[n] is also periodic with period 2N, 3N,...... The fundamental period N0 of x[n] is the smallest integer of N. A constant signal x[n] = 5 is periodic with any real integer, the fundamental period N0 = 1. Periodic Signals-DT 28  A discrete-time signal cos(ω0n), sin(ω0n), and ejω0n are periodic only if ω0/2π is an element of a rational value θ. 0  where 0  2 f is the radiant frequency 2    If ω0/2π is rational, then those signals are periodic. The samples fall at the same points in each super period of cos(ω0t), it may take several periods of cos(ω0t) to make one period of cos(ω0n). For ω0/2π to be rational, ω0 must contains the factor π. 0 0 m If  , then = , where m, N  S (set of integers) 2 2 N  If m/N is in reduced form, then m & N have no common factors and N is the fundamental period. Periodic Signals 29   The sum of two periodic signals may or may not be periodic. Consider two periodic signals x1(t) and x2(t) with fundamental period T1 and T2. x1 (t )  x1 (t  kT1 )   k & l are integers x2 (t )  x2 (t  lT2 )   If x(t) is periodic with a period T, we must have x(t )  1 x1 (t )   2 x2 (t ) 1 x1 (t  T )   2 x2 (t  T )  1 x1 (t  kT1 )   2 x2 (t  lT2 ) T  kT1  lT2 T1 l  T2 k Periodic Signals 30  Least common multiple (LCM)  LCM is also called the lowest common multiple or smallest common multiple of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b. Examples 1: Find the least common multiple for 4 and 6 Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36,... Multiples of 6 are: 6, 12, 18, 24, 30, 36,... Therefore, LCM(4, 6) = 12  Example 2: Find the least common multiple for 4, 6, and 8 Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36,... Multiples of 6 are: 6, 12, 18, 24, 30, 36,... Multiples of 8 are: 8, 16, 24, 32, 40,.... Therefore, LCM(4, 6, 8) = 24  Periodic Signals-CT 31  Example 1: x(t )  sin  2t  0 1 0  2  f 0   T0   2  Since  . Then the signal is periodic with afundamental period,T   Periodic Signals-CT 32  Example 2: x(t )  3cos (4t )  sin ( t ) 1 4 2  1  4  f1    T1  2 2  2 2  1 2    f 2     T2  2 2 2 2 T1  / 2        irrational value. T2 2 4 Therefore,the signal is aperiodic signal Periodic Signals-CT 33  Example 3: x(t )  cos (3 t )  2cos (4 t ) 1 3 3 2    T1  1  3  f1  2 2 2 3 2 4 1 2  4  f 2    2  T2  2 2 2 T1 2 / 3 4      rational value. T2 1/ 2 3 LCM  2 / 3,1/ 2   2 The signal is periodic with fundamental period, T  2 Periodic Signals-CT 34  Example 4: x(t )  cos(4t )  2sin(8t ) 1 4 2  1  4  f1     T1  2 2  2 2 8 4  2  8  f 2    T2  2 2  4 T1  / 2   2    rationa lvalue. T2  / 4 LCM  / 2,  / 4    / 2 Then the signal is periodic with a fundamental period, T   / 2 Periodic Signals-CT 35  Example 5:    x(t )  3sin 3 2t    7cos 6 2t 1 3 2 2 1  3 2  f1    T1  2 2 3 2  2 6 2 3 2  2  6 2  f 2     T2  2 2  3 2 m T1 2 / 3 2 2       rational value. k T2  / 3 2 1   LCM 2 / 3 2,  / 3 2  2 / 3 2 Thes ignal is periodic with fundamental period, T  2 / 3 2 OR T1k  T2 m  2 2  (1)  (2)  3 2 3 2 3 2 f  3 2 / 2 and the fundamental frequency,   2 f  3 2 Periodic Signals-DT 36  Example 1:  8  x[n]  cos  n  31  0 8 8 4 m 0   f0        rational value.. 31 2 62 31 N 0 The signal is periodic with fundamental period, N 0  31. Each m  4 periods of cos (0t ) make one period of cos (0 n) Periodic Signals-DT 37 n   Example 2: x[n]  cos   6 0 1 1 0   f 0     . 6 2 12 Hence,the signal is aperiodic.  Example 3: x[n]  sin  2n  x[n]  sin  2n  0 1 0  2  f 0     . 2  Hence,the signal is aperiodic. Periodic Signals-DT 38  Example 4: x[n]  1  sin 2  2 n  1 3 1 x[n]  1  sin  2 n   1  1  cos  4 n     cos  4 n . 2 2 2 Neglect the DCcurrent and consider the period of cos  4 n  only. 2 0 m 0  4  f 0  2 . 2 N0 Hence,the signal is periodic with fundamental period, N 0  1 Each m  2 periods of cos(0t ) make one period of cos(0 n) Periodic Signals-DT 39  Example 5:  6  x[n]  sin  n  1  7  0 3 6 0   f0   . 7 2 7 Hence,the signal is periodic with fundamental period, N 0  7. Each m  3 periods of sin(0t ) make one period of sin(0 n). Odd and Even Signals 40  An even signal is identical to its time reversed signal, i.e. it can be reflected in the y-axis and is equal to the original. x(t )  x(t )   Examples: x(t )  cos(t ) An odd signal is identical to its negated, time reversed signal, it must be symmetric around the origin at t = 0, i.e. it is equal to the negative reflected signal. x(t )   x(t )  Examples: x(t )  sin(t ) Odd and Even Signals 41  Any signal can be expressed as the sum of an odd signal and an even signal. 1 xe (t )   x(t )  x(t ) 2 1 xo (t )   x(t )  x(t ) 2 x(t )  xe (t )  xo (t ) Exponential and Sinusoidal Signals 42   Exponential and sinusoidal signals are characteristic of realworld signals and also from a basis for other signals. A generic complex exponential signal is of the form: x(t )  Ceat   where C and a are in general complex numbers. Lets investigate some special cases of this signal. Real exponential signals Exponential growth a0 C 0 Exponential decay a0 C 0 Periodic Complex Exponential & Sinusoidal Signals 43   Consider when a is purely imaginary: x(t )  Ce j0t By Euler’s relationship, this can be expressed as: cos(1) e j0t  cos 0t  j sin 0t  This is a periodic signals because: e j0 (t T )  cos 0 (t  T )  j sin 0 (t  T ) when    cos 0t  j sin 0t  e T  2 / 0 T0 = 2 /0 =  j0t 0  2 f 0 A closely related signal is the sinusoidal signal: x(t )  cos 0t    j ( t  ) A  t    A e cos Re     0 Always use: 0  A sin 0t     A Im e j (0t  )  T0 is the fundamental time period. w0 is the fundamental frequency. Exponential & Sinusoidal Signal Properties 44   Periodic signals, in particular complex periodic and sinusoidal signals, have infinite total energy but finite average power. Consider energy over one period: T0 2 E period   e j0t dt 0 T0   1dt  T0 Therefore: E   0  Average power: Pperiod   1 E period  1 T0 Useful to consider harmonic signals, terminology is consistent with its use in music, where each frequency is an integer multiple of a fundamental frequency General Complex Exponential Signals 45   So far, considered the real and periodic complex exponential Now consider when C is complex. Let us express C in polar form & a in C  C e j , a  r  j0 rectangular form: So Ceat  C e j e( r  j0 )t  C er t e j (0t  )  Using Euler’s relation Ceat  C er t cos((0   )t )  j C er t sin((0   )t )  These are damped sinusoids r 0 x  t   C er t cos(0t   ) r 0 Unit Step and Impulse Functions 46  Continuous Unit Step u(t) 1 ,t  0 u t    0 ,t  0  1 t Continuous Shifted Unit Step 1 ,t   u t     0 ,t   1 u(t- )  t Unit Step and Impulse Functions 47 Continuous Unit Step is discontinuous at t = 0, so is not differentiable! u(t)  Define delayed unit step:  1 1 ,t   / 2 t  u (t )  0 ,t   / 2    2 2 t 1   , otherwise  2   u(t) is continuous and differentiable u (t )  lim u (t )  0 1 ,  / 2  t   / 2 du (t )    dt  0 , otherwise Unit Impulse and Step Functions 48  The continuous unit impulse Function (Dirac delta) is defined: 0 t  0 x(t )   (t )    t  0    Note that it is discontinuous at t = 0 The arrow is used to denote area, rather than actual value Again, useful for an infinite basis t x(t )  u (t )    ( )d     (t )dt  1 - Signum Functions 49 Rectangular Functions 50 Triangular Functions 51 Cardinal Sine Functions 52 Linear Systems 53   A system takes a signal as an input and transforms it into another signal Linear systems play a crucial role in most areas of science     Closed form solutions often exist Theoretical analysis is considerably simplified Non-linear systems can often be regarded as linear, for small perturbations, so-called linearization For the remainder of the lecture/course we’re primarily going to be considering Linear Time Invariant systems (LTI) and consider their properties x(t) continuous time (CT) y(t) x[n] discrete time (DT) y[n] Examples of Simple Systems 54  To get some idea of typical systems (and their properties), consider the electrical circuit example: dvc (t ) 1 1  vc (t )  vs (t ) dt RC RC Note: this is a first order CT differential equation.  A discretised version of the electrical circuit is: The current through the capacitor RC k v[n]  v[n  1]  f [ n] RC  k RC  k  Example of second order system includes: d 2 y(t ) dy(t ) a  b  cy (t )  x(t ) 2 dt dt System described by order and parameters (a, b, c) i(t )  C dvc (t ) dt First Order Step Responses 55    People tend to visualise systems in terms of their responses to simple input signals. The dynamics of the output signal are determined by the dynamics of the system, if the input signal has no dynamics Consider when the input signal is a step at t, n = 1, y(0) = 0 First order CT differential system u(t) y(t) dy (t )  ay (t )  u (t  1) dt t System with and without Memory 56  A system is said to be memoryless if its output at any time depends on the input at that same time. y[n]  (2x[n]  x2 [n])2 , y (t )  2x (t ), y (t )  x 2 (t ), y (t )  te x (t )   e.g. a resistor is a memoryless CT system, where x(t) is current and y(t) is the voltage y  t   R x  t  Memory corresponds to a mechanism in the system that retains information about input values other than the current time. y[n]   k  x[k ]  x[n]  y[n  1]  x[n] n 1 1 t y  t    x   d C    e.g. DT system with memory is an accumulator (integrator) n y[n]   k  x[k ] and any system with a delay or advance is a memory system y[n]  x[n 1], y[n]  x[n  1] System with and without Memory 57  Examples: y(t )  x(t )  5 The system is memoryless because the value 5 is not affecting time function. y(t )  x(t  5) The system have memory because it is time shift, where the time function is affected. Let t = 0, then y(0) = x(5). y(t )  (t  5) x(t ) The system is memoryless because the value (t+5) is a scale and not affecting time function. y(t )  [ x(t  5)]2 The system have memory because it is time shift, where the time function is affected. System with and without Memory 58  Examples: y(t )  x(5) The system have memory because the output is a function of time regardless of input. Let t = 0, then y(0) = x(5). y(t )  x(2t ) The system have memory because it is time scaling, where the time function is affected. Let t = 1, then y(1) = x(2).  x(t )  x(t  2), t  0 y (t )   t  0, 0, The system have memory because the output y(t) is a function of x(t -2). Let t = 3, then y(3) = x(3) + x(1). Since y(3) depends on x(1), then the system have memory. Invertible and Inverse Systems 59   A system is said to be invertible if distinct inputs lead to distinct outputs (similar to matrix invertibility). If a system is invertible, an inverse system exists which, when cascaded with the original system, yields an output equal to the input of the first signal    E.g. the CT system is invertible: y  t   2 x  t  ; where w  t   0.5 y  t  w  t  recovers the original signal x  t  E.g. the CT & DT system is not-invertible: y  t   x2  t    are not  invertible y[n]  0   Widely used as a design principle:  Encryption, decryption, Encoder  System control, where the reference signal is input Invertible and Inverse Systems 60  Examples: y(t )  x(t  2) The system is invertible, since y(t) = x(t + 2) then y(t - 2) = x(t) and the inverse of the signal is y(t) = x(t - 2). Therefore the inverse of the function is z(t) = y(t - 2) = x((t - 2) + 2) = x(t). y(t )  x 2 (t ) The system is non-invertible because we can only determine the value of the input but not the sign of the input from the knowledge of the output. y(t )  x(2t ) The system is invertible, since y(t) = x(2t) then y(t/2) = x(t) and the inverse of the signal is y(t) = x(t/2). Therefore the inverse of the function is z(t) = y(t/2) =x(2t(1/2)) = x(t). System Causality 61  A system is causal if the output at any time depends only on values of the input at the present and past times. Referred to as non-anticipative, as the system output does not anticipate future values of the input. Most physical systems are causal.  E.g. The accumulator system is causal: n y[n]   k  x[k ] where y  n  only depends on x  n, x  n  1,.... time var ying function y t   x t     cos  t  1 E.g. The averaging/filtering system is non-causal y[n]  2 M1 1  k  M x[n  k ] where y  n depends on x  n  1, x  n  2 M y[n]  x[n]  x[n  1]  non  causal y[n]  x[n]  System Causality 62  Examples: y[n]  x[n]  x[n  1] The system is non-causal, when n = 0, y = x - x, the output at this time depends on a future value of the input. y(t )  x(t  1) The system is non-causal, when t = 0, y(0) = x(1) - x, the output at this time depends on a future value of the input. y[n]  x[n] The system is non-causal, since when n < 0, e.g. n = -1, we see that y[-1] = x, so the output at this time depends on a future value of the input. y(t )  x(t )cos(t  1) The system is causal because the output at any time equals the input at the same time multiplied by a number that varies with time cos(t +1). System Causality 63  Examples: y(t0 )   t0  a  x(t )dt The system is non-causal, since we do not know the value of a. If a > 0, then the output depends on a future value of input.  x(t )  x(t  2), t  0 y (t )   t0 0, The system is causal, when t < 0, the output does not depend on the input. When t ≥ 0, the output y(t) depends on current input x(t) and past input x(t - 2). The system is causal because the output does not depend on a future input. System Stability 64  Informally, a stable system is one in which small input signals lead to responses that do not diverge. If an input signal is bounded, then the output signal must also be bounded, if the system is stable x : x  U y  V  To show a system is stable, we have to do it for all input signals.  E.g. Consider the DT system of the bank account y (t )  e x (t ) stable y (t )  tx(t ) unstable   y[n]  x[n]  1.01y[n  1] when x[n]   (t ), y  0 unstable: it grows without bound, due to 1.01 multiplier. E.g. the CT, is stable if RC >0, because it dissipates energy dvc (t ) 1 1  vc (t )  vs (t ) dt RC RC System Stability 65  Examples: y(t )  tx(t ) The system is not stable, when a constant input x(t) = 1, yields an output y(t) = t, which is not bounded - no matter what finite constant we pick, y(t) will exceed the constant for some t. y(t )  e x (t ) The system is stable, assume the input is bounded x(t )  B, or  B  x(t )  B for all t , then y(t ) is bounded e B  y(t )  e B. System Stability 66  Examples:  x(t )  x(t  2), t  0 y (t )   t  0, 0, The system is stable, assume the input is bounded B  0 and | x(t ) | B  t. When t  0 y (t )  0, so | y (t ) | 0  2 B. When t  0, (| y (t ) || x(t )  x(t  2) || x(t ) |  | x(t  2) | B  B  2 B). So | y (t ) | 2 B  t y (t ) is bounded. Since y (t ) is bounded, then the system is stable. System Stability 67  Examples: dV (t ) i (t )  C c dt Let i (t )  B1u (t ), where B1  0 1 t 1 t 1 t B1 Vc (t )   i ( )d   B1u ( )d   B1d  t. C  C  C 0 C B1 Vc (t )  tgrows linearly with t and as t ,Vc (t ) . C Bounded-input gives unbounded-output, so a capacitor is not BIBO stable. Time Invariance 68   A system is time invariant if its behavior and characteristics are fixed over time. We would expect to get the same results from an input-output experiment, if the same input signal is fed in at a different time.  E.g. The following CT system is time-invariant y(t )  sin( x(t )) y1 (t  t0 )  sin( x1 (t  t0 )) because it is time-invariant to a time shift, let x2(t) = x1(t - t0) y2 (t )  sin( x2 (t ))  sin( x1 (t  t0 ))  y1 (t  t0 )  E.g. The following DT system is time-varying y[n]  nx[n]  x1[n]   [n]  y1[n]  0 x2 [n]   [n  1]  y2 [n]   [n  1] Because the system parameter that multiplies the input signal is time varying. Time Invariance 69 Examples of time-invariant systems: RC circuitis, since R and C do not change in time. y (t )  sin( x(t )) y (t )  3x(t  2) y (t )  x(t )  x(t  1) y (t )  x 2 (t ) Examples of time-variant systems: y (t )  0 The system produces zero output sequence for any input sequence. y (t )  tx(t ) The system depends on what time it is. y (t )  t  x(t  1) The system depends on time. Time Invariance 70  Examples 1: y (t )  x(2t ) Consider an arbitrary input x1 (t ), then the resulting output y1 (t ) is y1 (t )  x1 (2t ) Consider a time shift delay of y (t ) by 2 y1 (t  2)  x1 (2t  2)....................................................... (1) Consider a second input x2 (t ) obtained by delaying x1 (t ) by 2 x2 (t )  x1 (t  2) Then the resulting second output y2 (t ) is y2 (t )  x2 (2t )  x1 (2t  4)................................................ (2) As shown from 1 & 2, y1 (t  2)  y2 (t ), so the system is time-varying. Time Invariance 71  Examples 1: y(t )  x(2t ) Time Invariance 72  Examples 2: y (t )  tx(t ) Consider an arbitrary input x1 (t ), then the resulting output y1 (t ) is y1 (t )  tx1 (t ) Consider a time shift delay of y1 (t ) by 2 y1 (t  2)  (t  2) x1 (t  2)............................................. (1) Consider a second input x2 (t ) obtained by delaying x1 (t ) by 2 x2 (t )  x1 (t  2) Then the resulting second output y2 (t ) is y2 (t )  tx2 (t )  tx1 (t  2).................................................. (2) As shown from 1 & 2, y1 (t  2)  y2 (t ), so the system is time-varying. Time Invariance 73  Examples 2: y(t )  tx(t ) Time Invariance 74  Examples 3: y (t )  x 2 (t ) Consider an arbitrary input x1 (t ), then the resulting output y1 (t ) is y1 (t )  x12 (t ) Consider a time shift delay of y1 (t ) by 2 y1 (t  2)  x12 (t  2).......................................................... (1) Consider a second input x2 (t ) obtained by delaying x1 (t ) by 2 x2 (t )  x1 (t  2) Then the resulting second output y2 (t ) is y2 (t )  x2 2 (t )  x12 (t  2).................................................. (2) As shown from 1 & 2, y1 (t  2)  y2 (t ), so the system is time-invariant. Time Invariance 75  Examples 3: y(t )  x 2 (t ) Time Invariance 76  Examples 4:  x(t )  x(t  2), t  0 y (t )   t  0, 0, Consider an arbitrary input x1 (t ), then the resulting output y1 (t ) is y1 (t )  x1 (t )  x1 (t  2) Consider a time shift delay of y1 (t ) by 2 y1 (t  2)  x1 (t  2)  x1 (t  4)............................................. (1) Consider a second input x2 (t ) obtained by delaying x1 (t ) by 2 x2 (t )  x1 (t  2) Then the resulting second output y2 (t ) is y2 (t )  x2 (t )  x2 (t  2)  x1 (t  2)  x1 (t  4)...................... (2) As shown from 1 & 2, y1 (t  2)  y2 (t ), so the system is time-invariant. Time Invariance 77  Examples 4:  x(t )  x(t  2), t  0 y (t )   t  0, 0, Linearity 78   The most important property that a system possesses is linearity. It means allows any system response to be analysed as the sum of simpler responses (convolution). Simplistically, it can be imagined as a line. y x A linear system must satisfy the two properties: Additivity : Given that Tx1 (t )  y1 (t ) and Tx2 (t )  y2 (t ), then T  x1 (t )  x2 (t )   y1 (t )  y2 (t ), for any input signal x1 (t ) and x2 (t ) Scaling : T  x(t )   y (t ), for any input signal x(t ) and any scalar  Superposition : T 1 x1 (t )   2 x2 (t )   1 y1 (t )   2 y2 (t )   Linear: y(t) = 3*x(t) Non-linear: y(t) = 3*x(t) + 2, why? y(t) = 3*x2(t) why? Linearity 79  Suppose an input signal x[n] is made of a linear sum of other (basis/simpler) signals xk[n]: x[n]   k ak xk [n]  a1 x1[ n]  a2 x2[ n]  a3 x3[ n]    then the (linear) system response is: y[n]   k ak yk [n]  a1 y1[ n]  a2 y2[ n]  a3 y3[ n]     The basic idea is that if we understand how simple signals get affected by the system, we can work out how complex signals are affected, by expanding them as a linear sum This is known as the superposition property which is true for linear systems in both CT & DT Linearity 80  Examples 1: y (t )  x(2t ) Consider an arbitrary 2 inputs x1 (t ) and x2 (t ) x1 (t )  y1 (t )  x1 (2t ), x2 (t )  y2 (t )  x2 (2t ) We obtain the sum of the 2 outputs 1 y1 (t )   2 y2 (t )  1 x1 (2t )   2 x2 (2t ) E Q 1 Consider athird input x3 (t ) obtained by combining x1 (t ) and x2 (t ) x3 (t )  1 x1 (t )   2 x2 (t ) Then the third output y3 (t ) is EQ2 y3 (t )  x3 (2t )  1 x1 (2t )   2 x2 (2t )  1 y1 (t )   2 y2 (t ) Therefor, the system is linear. Linearity 81  Examples 2: y (t )  tx(t ) Consider an arbitrary 2 inputs x1 (t ) and x2 (t ) x1 (t )  y1 (t )  tx1 (t ), x2 (t )  y2 (t )  tx2 (t ) We obtain the sum of the 2 outputs 1 y1 (t )   2 y2 (t )  1tx1 (t )   2tx2 (t ) EQ1 Consider athird input x3 (t ) obtained by combining x1 (t ) and x2 (t ) x3 (t )  1 x1 (t )   2 x2 (t ) Then the third output y3 (t ) is y3 (t )  tx3 (t )  t 1 x1 (t )   2 x2 (t ) ΕΘ2  1tx1 (t )   2tx2 (t )  1 y1 (t )   2 y2 (t ) Therefor, the system is linear. Linearity 82  Examples 3: 2 y (t )  x (t ) Consider an arbitrary 2 inputs x1 (t ) and x2 (t ) x1 (t )  y1 (t )  x12 (t ), x2 (t )  y2 (t )  x2 2 (t ) We obtain the sum of the 2 outputs 1 y1 (t )   2 y2 (t )  1 x12 (t )   2 x2 2 (t ) EQ1 Consider a third input x3 (t ) obtained by combining x1 (t ) and x2 (t ) x3 (t )  1 x1 (t )   2 x2 (t ) Then the third output y3 (t ) is y3 (t )  x32 (t )  1 x1 (t )   2 x2 (t )  2  12 x12 (t )  21 x1 (t ) 2 x2 (t )   2 2 x2 2 (t ) EQ2 Since y3 (t )  1 x12 (t )   2 x2 2 (t ). Then, the system is nonlinear. Linearity 83  Examples 4:  x(t )  x(t  2), t  0 y (t )   t  0, 0, Consider an arbitrary 2 inputs x1 (t ) and x2 (t ) x1 (t )  y1 (t )  x1 (t )  x1 (t  2), x2 (t )  y2 (t )  x2 (t )  x2 (t  2) We obtain the sum of the 2 outputs 1 y1 (t )   2 y2 (t )  1  x1 (t )  x1 (t  2)    2  x2 (t )  x2 (t  2) EQ1 Consider a third input x3 (t ) obtained by combining x1 (t ) and x2 (t ) x3 (t )  1 x1 (t )   2 x2 (t ) Then the third output y3 (t ) is y3 (t )  x3 (t )  x3 (t  2)  1 x1 (t )   2 x2 (t )  1 x1 (t  2)   2 x2 (t  2)  1[ x1 (t )  x1 (t  2)]   2 [ x2 (t )  x2 (t  2)] EQ2  1 y1 (t )   2 y2 (t ), Therefor, the system is linear. System Structures 84   Systems are generally composed of components (sub-systems). We can use our understanding of the components and their interconnection to understand the operation and behavior of the y overall system. x System 1 System 2 Series/cascade Parallel System 1 x y + System 2 Feedback x + System 1 System 2 y Next Lecture Fourier Series