Brief Review Of Discrete-Time Signal Processing Brief Review Of Random .
Chapter 1 Brief Review of Discrete-Time Signal Processing Brief Review of Random Processes References: A.V.Oppenheim and A.S.Willsky, Signals and Systems, Prentice Hall, 1996 J.G.Proakis and D.G.Manolakis, Introduction to Digital Signal Processing, Macmillan, 1988 A.V.Oppenheim and R.W.Schafer, Discrete-Time Signal Processing, Prentice Hall, 1998 P.M.Clarkson, Optimal and Adaptive Signal Processing, CRC, 1993 P.Stoica and R.Moses, Introduction to Spectral Analysis, Prentice Hall, 1997 1
Brief Review of Discrete-Time Signal Processing There are 3 types of signals that are functions of time: continuous-time (analog) : defined on a continuous range of time discrete-time : defined only at discrete instants of time ( ,(n-1)T,nT, (n 1)T, ) digital (quantized) : both time and amplitude are discrete x(t) analog signal digital signal quantized processor signal x(nT) xQ(nT) T sampled signal Time & Amplitude Time discrete Time & Amplitude continuous Amplitude continuous discrete 2
Digital Signal Processing Applications Speech Coding (compression) Synthesis (production of speech signals, e.g., speech development kit by Microsoft ) Recognition (e.g., PCCW’s 1083 telephone number enquiry system and many applications for disabled persons as well as security) Animal sound analysis Music Generation of music by different musical instruments such as piano, cello, guitar and flute using computer Song with low-cost electronic piano keyboard quality 3
Image Compression Recognition such as face, palm and fingerprint Construction of 3D objects from 2D images Animation, e.g., “Toy Story (反斗奇兵)” Special effects such as adding Forrest Gump to a film of President Nixon in “阿甘正傳” and removing some objects in a photograph or movie Digital Communications Encryption Transmission and Reception (coding / decoding, modulation / demodulation, equalization) Biometrics and Bioinformatics Digital Control 4
Transform from Time to Frequency x(t ) transform X (ω) inverse transform Fourier Series express periodic signals using harmonically related sinusoids different definitions for continuous-time & discrete-time signals frequency ω takes discrete values: ω0 , 2ω0 , 3ω0 , . Fourier Transform frequency analysis tool for aperiodic signals defined on a continuous range of ω different definitions for continuous-time & discrete-time signals Fast Fourier transform (FFT) – an computationally efficient method for computing Fourier transform of discrete signals 5
6
Transform Time Domain Frequency Domain periodic & continuous aperiodic & discrete ω0 T P / 2 jω 0 kt ck x ( t ) e dt , 2π T P / 2 Fourier Series x(t ) ck e jkω0 t , k ω0 2π / TP Fourier Transform Discrete-Time Fourier Transform aperiodic & continuous 1 j ωt x(t ) X (ω)e dω 2π aperiodic & discrete T π/T jωnT x(nT ) dω , X (ω)e 2π π / T T is the sampling interval periodic & discrete N 1 Discrete(-Time) j 2 πkn / N x ( n ) c e , k Fourier Series k 0 TP N and T 1 7 TP is the period aperiodic & continuous X (ω) x(t )e jωt dt periodic & continuous X (ω) x(nT )e jωnT n periodic & discrete 1 N 1 j 2 πkn / N ck x ( n )e N n 0
Fourier Series Fourier series are used to represent the frequency contents of a periodic and continuous-time signal. A continuous-time function x(t ) is said to be periodic if there exists TP 0 such that x(t ) x(t TP ), t ( , ) (I.1) The smallest TP for which (I.1) holds is called the fundamental period. Every periodic function can be expanded into a Fourier series as x(t ) c k e jkω0t , where k t ( , ) (I.2) ω 0 TP / 2 jω kt ck x(t )e 0 dt 2π TP / 2 (I.3) and ω0 2π / TP is called the fundamental frequency. 8
Example 1.1 The signal x(t ) cos(100πt ) cos(200πt ) is a periodic and continuoustime signal. The fundamental frequency is ω0 100π . The fundamental period is then TP 2π /(100π) 1 / 50 : 1 1 1 x t cos 100π t cos 200π t 50 50 50 cos(100πt 2π ) cos(200πt 4π ) cos(100πt ) cos(200πt ) x(t ) e jω 0 t e jω 0 t e j 2 ω 0 t e j 2 ω 0 t Since x(t ) cos(100πt ) cos(200πt ) 2 2 By inspection and using (I.2), we have c1 1 / 2 , c 1 1 / 2 , c2 1 / 2 , c 2 1 / 2 while all other Fourier series coefficients are equal to zero. 9
Fourier Transform Fourier transform is used to represent the frequency contents of an aperiodic and continuous-time signal x(t ) : Forward transform: X (ω) x(t )e jωt dt (I.4) and Inverse transform: 1 j ωt x(t ) X ( ω ) e dω 2π (I.5) Some points to note: Fourier spectrum (both magnitude and phase) are continuous in frequency and aperiodic Convolution in time domain corresponds to multiplication in Fourier transform domain, i.e., x(t ) y (t ) X (ω) Y (ω) 10
Example 1.2 Find the Fourier transform of the following rectangular pulse: 1, x(t ) 0, Using (I.4), X (ω) T1 jωt dt T1 e 11 t T1 t T1 2 sin(ωT1 ) ω
Example 1.3 Find the inverse Fourier transform of 1, X (ω) 0, Using (I.5), ω W ω W 1 W jω t sin(Wt ) x(t ) W e dω 2π πt 12
Discrete-Time Fourier Transform (DTFT) DTFT is a frequency analysis tool for aperiodic and discrete-time signals. If we sample an aperiodic and continuous-time function x(t ) with a sampling interval T , the sampled output x s (t ) is expressed as x s (t ) x(t ) δ(t nT ) (I.6) n 13
The DTFT can be obtained by substituting x s (t ) into the Fourier transform equation of (1.4): X (ω) xs (t )e jωt dt n x(t ) δ(t nT )e jωt dt x(t )δ(t nT )e jωt dt n x(nT )e jωnT (I.7) n where sifting property of unit-impulse function is employed to obtain (1.7): f (t )δ(t t 0 )dt f (t 0 ) 14
Some points to note: DTFT spectrum (both magnitude and phase) is continuous in frequency and periodic with period 2π / T When the sampling interval is normalized to 1, we have Forward Transform: and Inverse Transform: X (ω) x(n)e jωn (I.8) 1 π jωn x ( n) X (ω)e dω 2π π (I.9) n Discrete-Time Fourier Series (DTFS) DTFS is used for analyzing discrete-time periodic signals. It can be derived from the Fourier series. 15
Example 1.4 Find the DTFT of the following discrete-time signal: 1, x[n] 0, n N1 n N1 Using (I.8), X (ω) N1 e N1 2 jω n n N1 e jωN1 (1 e jω e j 2ω L e j 2 N1ω 16 sin(( N1 1 2)ω) ) sin(ω 2)
z-Transform It is a useful transform of processing discrete-time signal. In fact, it is a generalization of DTFT for discrete-time signals X ( z ) Z {x[n]} x[n]z n n (I.10) where z is a complex variable. Substituting z e jω yields DTFT. Moreover, substituting z re jω gives X ( z ) x[n]r n e jnω F {x[n]r n } n 17 (I.11)
Advantages of using z -transform over DTFT: can encompass a broader class of signal since Fourier transform does not converge for all sequences: A sufficient condition for convergence of the DTFT is X (ω) x(n) e j ωn n x(n) n (I.12) Therefore, if x(n) is absolutely summable, then X (ω) exists. On the other hand, by representing z re jω , the z -transform exists if X ( z ) X (re jω ) x(n)r n n e jωn x(n)r n n (I.13) we can choose a region of convergence (ROC) for z such that the z transform converges notation convenience : z e jω can solve problems in discrete-time signals and systems, e.g. difference equations 18
Example 1.5 Determine the z-transform of x[n] a n u[n]. a X ( z) n u[n]z n n X (z ) converges if az 1 n n 0 (az 1 ) n n 0 . This requires az 1 1 or z a , and 1 X ( z) 1 az 1 Notice that for another signal x[n] a n u[ n 1] , 1 n X ( z ) ( a ) z n n a m 1 19 m m z ( ) 1 a z m 1 m
In this case, X (z ) converges if a 1 z 1 or z a , and 1 X ( z) 1 az 1 ROC of ROC of x[n] a n u[n] x[n] a n u[ n 1] Some points to note: Different signals can give same z-transform, although the ROCs differ When x[n] a n u[n] with a 1, its DTFT does not exist 20
21
Transfer Function and Difference Equation A linear time-invariant (LTI) system with input sequence x(n) and output sequence y (n) are related via an Nth-order linear constant coefficient difference equation of the form: N M k 0 k 0 a k y (n k ) bk x(n k ), a 0 0, b0 0 (I.14) Applying z -transform to both sides with the use of the linearity property and time-shifting property, we have N ak z k 0 k M Y ( z ) bk z k X ( z ) k 0 (I.15) The system (or filter) transfer function is expressed as M Y ( z) H ( z) X ( z) bk z k 0 N ak z k 0 1 k k M 1 (1 c k z ) b0 k 1 N a 0 (1 d z 1 ) k (I.16) k 1 where each (1 ck z ) contributes a zero at z ck and a pole at z 0 while each (1 d k z 1 ) contributes a pole at z d k and a zero at z 0 . 22
The frequency response of the system or filter can be computed as H (ω) H ( z ) z exp( jω) (I.17) From (1.14), the output y (n) is expressed as N 1 M y ( n ) bk x ( n k ) a k y ( n k ) a 0 k 0 k 1 (I.18) When at least one of the {a1 , a 2 , L , a N } is non-zero, then y (n) depends on its past samples as well as the input signal x(n) . The system or filter in this case is known as an infinite impulse response (IIR) system. Applying inverse DTFT or z transform to the transfer function, it can be shown that the system impulse response is of infinite duration. When all {a1 , a 2 , L , a N } are equal to zero, y (n) depends on x(n) only. It is known as a finite impulse response (FIR) system because the impulse response is of finite duration. 23
Example 1.6 Consider a LTI system with the input x[n] and output y[n] satisfy the following linear constant-coefficient difference equation, 1 1 y[n] y[n 1] x[n] x[n 1] 2 3 Find the system function and frequency response. Taking z-transform on both sides, Thus, 1 1 Y ( z ) z 1Y ( z ) X ( z ) z 1 X ( z ) 2 3 1 jω 1 1 1 e 1 z Y ( z) 3 3 and H (ω) H ( z ) z exp( jω) H ( z) 1 jω X ( z ) 1 1 z 1 1 e 2 2 24
25
Example 1.7 Suppose you need to high-pass the signal x[n] by the high-pass filter with the following transfer function H ( z) 1 1 0.99 z 1 How to obtain the filtered signal y[n] ? Y ( z) 1 1 H ( z) Y ( z ) 0 . 99 z Y ( z) X ( z) 1 X ( z ) 1 0.99 z Taking the inverse z-transform y[n] 0.99 y[n 1] x[n] y[n] 0.99 y[n 1] x[n] ( y[ 1] 0 for initialization) 26
27
Causality, Stability and ROC: h[n] 0 for all n 0 , Causality condition: h[n] is right-sided The ROC for H (z ) is the exterior of an origin-centered circle (including z ) If H ( z ) is rational, the ROC for H ( z ) is the exterior outside the outermost pole. Stability condition: h[n] n H (e jω ) , i.e., the Fourier transform of h[n], converges The ROC for H ( z ) includes the unit circle z 1 28
Example 1.8 Verify if the system impulse response h[n] 0.5 n u[n] is causal and stable. It is obvious that h[n] is causal because h[n] 0 for all n 0 . On the other hand, n H ( z ) 0.5 u[n]z n H (z ) converges if 0.5 z n 0 1 n n 1 n ( 0 .5 z ) n 0 1 1 0.5 z 1 . This requires 0.5 z 1 1 or z 0.5 , i.e., ROC for H (z ) is the exterior outside the pole of 0.5 (Notice that for another impulse response h[n] 0.5n u[ n 1], and it corresponds to an unstable system because the ROC for H ( z ) is z 0 .5 ) 29
The z-transform for h[n] is H ( z) 1 1 0 .5 z 1 , z 0.5 Hence it is stable because the ROC for H (z ) includes the unit circle z 1 On the other hand, its stability can also be shown using: n n 0 n 2 3 h[n] 0.5 1 0.5 0.5 L 1 2 1 0.5 30
Brief Review of Random Processes Basically there are two types of signals: Deterministic Signals exactly specified according to some mathematical formulae characterized by finite parameters e.g., exponential signal, sinusoidal signal, ramp signal, etc. a simple mathematical model of a musical signal is x(t ) a (t ) cm cos(2πmf 0t φ m ) where: m 1 f 0 is the fundamental frequency or pitch cm is the amplitude and φ m is the phase of the mth harmonic 31
a (t ) is the envelope 32
33
Random Signals cannot be directly generated by any formulae and their values cannot be predicted characterized by probability density function (PDF), mean, variance, power spectrum, etc. , stock values, autoregressive (AR) process, e.g., thermal noise moving average (MA) process, etc. a simple voiced discrete-time speech model is P x[n] ai x[n i ] w[n] where i 1 { ai } are called the AR parameters w[n] is a noise-like process P is the order of the AR process 34
Definitions and Notations 1. Mean Value The mean value of a real random variable x(n) at time n is defined as µ(n) E{x(n)} x(n) f ( x(n))d ( x(n) ) (I.19) where f ( x(n)) is the PDF of x(n) such that f ( x(n))d ( x(n) ) 1 and f ( x(n)) 0 Note that, in general, and µ(m) µ(n), m n 1 N 1 µ ( m) x ( n) N n 0 The mean value is also called expected value and ensemble mean. 35 (I.20) (I.21)
2. Moment Moment is the generalization of the mean value: E{( x(n) ) } ( x(n) )m f ( x( n))d ( x( n) ) m (I.22) When m 1, it is the mean while when m 2 , it is called the mean square value of x(n) . 3. Variance The variance of a real random variable x(n) at time n is defined as σ (n) E{( x(n) µ(n)) } ( x(n) µ(n)) 2 f ( x(n))d ( x(n) ) 2 2 It is also called second central moment. 36 (I.23)
Example 1.9 Determine the mean, second-order moment, variance of a quantization error, x , with the following PDF: a a µ x f ( x)dx x a 1 1 1 2 dx x 0 2a 2a 2 a a a2 1 1 3 2 2 2 1 x E{x } x f ( x)dx x dx 2a 2a 3 a 3 a 2 a σ 2 E{( x µ) 2 } E{x 2 } 3 a 37
4. Autocorrelation The autocorrelation of a real random signal x(n) is defined as R xx (m, n) E{x(m) x(n)} x(m) x(n) f ( x(m), x(n) )d ( x(m) )d ( x(n) ) (I.24) where f ( x(m), x( n) ) is the joint PDF of x(m) and x(n) . It measures the degree of association or dependence between x at time index n and at index m . In particular, R xx (n, n) E{x 2 ( n)} (I.25) is the mean square value or average power of x(n) . Moreover, when x(n) has zero-mean, then σ 2 ( n) R xx (n, n) E{x 2 ( n)} That is, the power of x(n) is equal to the variance of x(n) . 38 (I.26)
5. Covariance The covariance of a real random signal x(n) is defined as C xx (m, n) E{( x(m) µ(m) )( x( n) µ(n) )} Expanding (I.27) gives In particular, C xx (m, n) E{x(m) x(n)} µ( m)µ(n) C xx (n, n) E{( x(n) µ(n) )2 } σ 2 (n) is the variance, and for zero-mean x(n) , we have C xx (m, n) R xx (m, n) 39 (I.27)
6. Crosscorrelation The crosscorrelation of two real random signals x(n) and y (n) is defined as R xy (m, n) E{x(m) y (n)} x(m) y (n) f ( x(m), y (n) )d ( x(m) )d ( y (n) ) (I.28) where f ( x(m), y (n) ) is the joint PDF of x(m) and y (n) . It measures the correlation of x(n) and y (n) . The signals x(m) and y (n) are uncorrelated if Rxy (m, n) E{x(m)} E{x(n)}. 7. Independence Two real random variables x(n) and y (n) are said to be independent if f ( x(n), y (n) ) f ( x(n) ) f ( y (n) ) E{x(n) y (n)} E{x(n)} E{ y (n)} Q.: Does “uncorrelated” implies “independent” or vice versa? 40 (I.29)
8. Stationarity A discrete random signal is said to be strictly stationary if its k -th order PDF f ( x(n1 ), x(n2 ), L , x(nk )) is shift-invariant for any set of n1 , n2, L , nk and for any k . That is f ( x(n1 ), x(n2 ), L , x(nk )) f ( x(n1 n0 ), x(n2 n0 ), L , x(nk n0 )) (I.30) where n0 is an arbitrary shift and for all k . In particular, a real random signal is said to be wide-sense stationary (WSS) if the first and second order moments, viz., its mean and autocorrelation, are shift-invariant. This means µ E{x(n)} E{x(m)}, m n (I.31) and R xx (i ) R xx (m n) R xx (m, n) E{x(m) x(n)} where i m n is called the correlation lag. 41 (I.32)
Three important properties of R xx (i ) : (i) R xx (i ) is an even sequence, i.e., R xx (i ) R xx ( i ) (I.33) and hence is symmetric about the origin. Q.: Why is it an even sequence? (ii)The mean square value or power is greater than or equal the magnitude of the correlation for any other lag, i.e., E{x 2 (n)} R xx (0) R xx (i ) , i 0 (I.34) which can be proved by the Cauchy-Schwarz inequality: E{a b} E{a 2 } E{b 2 } (iii)When x(n) has zero-mean, then σ 2 E{x 2 (n)} R xx (0) 42 (I.35)
9. Ergodicity A stationary process is said to be ergodic if its time average using infinite samples equals its ensemble average. That is, the statistical properties of the process can be determined by time averaging over a single sample function of the process. For example, Ergodic in the mean if 1 N / 2 1 µ E{x(n)} lim x ( n) N N n N / 2 Ergodic in the autocorrelation function if 1 N / 2 1 R xx (i ) E{x(n) x(n i )} lim x ( n) x ( n i ) N N n N / 2 Unless stated otherwise, we assume that random signals are ergodic (and thus stationary) in this course. 43
Example 1.10 Consider an ergodic stationary process { x[n] }, L, 1,0,1,L which is uniformly distributed between 0 and 1. The ensemble average or mean of x[n] at time m is 1 1 1 1 µ[m] x[m] f ( x[m])dx[m] x[m]dx[m] x 2 [m] 2 2 0 0 It is clear that the mean of x[n] is also µ 0.5 for all n Because of ergodicity, the time average is 1 lim N N N / 2 1 1 x[n] µ 2 n N / 2 44
10. Power Spectrum For random signals, power spectrum or power spectral density (PSD) is used to describe the frequency spectrum. Q.: Can we use DTFT to analyze the spectrum of random signal? Why? The PSD is defined as: Φ xx (ω) R xx (i )e jωi Z [R xx (i )] z exp( jω) i (I.36) Given Φ xx (ω) , we can get R xx (i ) using 1 π jωi R xx (i ) Φ xx (ω)e dω 2π π Q.: Why? 45 (I.37)
Under a mild assumption: 1 N lim k R xx (k ) 0 N N k N it can be proved (1.36) is equivalent to 1 N 1 j ωn Φ xx (ω) lim E x ( n)e N N n 0 2 (1.38) N 1 Since x(n)e jωn corresponds to the DTFT of x(n) , we can consider the n 0 2 PSD as the time average of X (ω) based on infinite samples. (1.38) also implies that the PSD is a measure of the mean value of the DTFT of x(n) . 46
Common Random Signal Models 1. White Process A discrete-time zero-mean signal w(n) is said to be white if σ 2w , m n R ww (m n) E{w(n) w(m)} 0, otherwise Moreover, the PSD of w(n) is flat for all frequencies: (I.39) Φ ww (ω) Rww (i )e jωi R ww (0) e jω 0 σ 2w i Notice that white process does not specify its PDF. They can be of Gaussian-distributed, uniform-distributed, etc. 47
2. Autoregressive Process An autoregressive (AR) process of order M is defined as x(n) a1 x(n 1) a 2 x(n 2) L a M x(n M ) w(n) where w(n) is a white process. Taking the z -transform of (1.40) yields H ( z) 1 X ( z) W ( z ) 1 a1 z 1 a 2 z 2 L a M z M Let h(n) Z 1 {H ( z )}, we can write k k x(n) h(n) w(n) h(n k ) w(k ) w(n k )h(k ) Q.: What Is the mean value of x(n) ? 48 (I.40)
Input-output relationship of random signals is: R xx (m) E{x(n) x(n m)} E h(k1 ) w(n k1 ) h(k 2 ) w(n m k 2 ) k1 k2 h(k1 )h(k 2 ) E{w(n k1 ) w(n m k 2 )} k1 k2 h(k1 )h(k 2 ) Rww (m k1 k 2 ) k1 k2 k k1 Rww (m k ) h(k1 )h(k k1 ), R xx (m) Rww (m) g (m), Φ xx (ω) Φ ww (ω) G (ω), g (k ) k k 2 k1 h(k1 )h(k k1 ) h(k ) h( k ) k1 G (ω) H (ω) Φ xx (ω) Φ ww (ω) H (ω) 49 2 2 (I.41)
Note that (1.41) applies for all stationary input processes and impulse responses. In particular, for the AR process, we have Φ xx (ω) σ 2w 1 a1e jω a 2 e j 2ω L a M e jMω 2 (1.42) 3. Moving Average Process A moving average (MA) process of order N is defined as x(n) b0 w(n) b1w(n 1) L b N w(n N ) (I.43) Applying (1.41) gives Φ xx (ω) b0 b1e jω L bN e 50 j ωN 2 σ 2w (I.44)
4. Autoregressive Moving Average Process An autoregressive moving average (ARMA) process is defined as x(n) a1 x(n 1) a 2 x(n 2) L a M x(n M ) b0 w(n) b1 w(n 1) L b N w(n N ) (I.45) Applying (1.41) gives Φ xx (ω) b0 b1e jω L bN e jNω 2 1 a1e jω a 2 e j 2ω L a M e 51 jMω 2 σ 2w (1.46)
Questions for Discussion 1. Consider a signal x(n) and a stable system with transfer function H ( z ) B ( z ) / A( z ) . Let the system output with input x(n) be y (n) . Can we always recover x(n) from y (n) ? Why? You may consider the B( z ) 1 2 z 1 B( z ) 1 0.5 z 1 and A( z ) 1. simple cases of and A( z ) 1 as well as 2. Given a random variable x with mean µ x and variance σ 2x . Determine the mean, variance, mean square value of y ax b where a and b are finite constants. 3. Is AR process really stationary? You can answer this question by examining the autocorrelation function of a first-order AR process, say, x(n) ax(n 1) w(n) 52
Fourier Series Fourier series are used to represent the frequency contents of a periodic and continuous-time signal. A continuous-time function x(t) is said to be periodic if there exists TP 0 such that x(t) x(t TP), t ( , ) (I.1) The smallest TP for which (I.1) holds is called the fundamental period. Every periodic function can be expanded into a Fourier series as
2.1 Sampling and discrete time systems 10 Discrete time systems are systems whose inputs and outputs are discrete time signals. Due to this interplay of continuous and discrete components, we can observe two discrete time systems in Figure 2, i.e., systems whose input and output are both discrete time signals.
2.1 Discrete-time Signals: Sequences Continuous-time signal - Defined along a continuum of times: x(t) Continuous-time system - Operates on and produces continuous-time signals. Discrete-time signal - Defined at discrete times: x[n] Discrete-time system - Operates on and produces discrete-time signals. x(t) y(t) H (s) D/A Digital filter .
6 POWER ELECTRONICS SEGMENTS INCLUDED IN THIS REPORT By device type SiC Silicon GaN-on-Si Diodes (discrete or rectifier bridge) MOSFET (discrete or module) IGBT (discrete or module) Thyristors (discrete) Bipolar (discrete or module) Power management Power HEMT (discrete, SiP, SoC) Diodes (discrete or hybrid module)
Definition and descriptions: discrete-time and discrete-valued signals (i.e. discrete -time signals taking on values from a finite set of possible values), Note: sampling, quatizing and coding process i.e. process of analogue-to-digital conversion. Discrete-time signals: Definition and descriptions: defined only at discrete
Computation and a discrete worldview go hand-in-hand. Computer data is discrete (all stored as bits no matter what the data is). Time on a computer occurs in discrete steps (clock ticks), etc. Because we work almost solely with discrete values, it makes since that
What is Discrete Mathematics? Discrete mathematics is the part of mathematics devoted to the study of discrete (as opposed to continuous) objects. Calculus deals with continuous objects and is not part of discrete mathematics. Examples of discrete objects: integers, distinct paths to travel from point A
Discrete Event Simulation (DES) 9 Tecniche di programmazione A.A. 2019/2020 Discrete event simulation is dynamic and discrete It can be either deterministic or stochastic Changes in state of the model occur at discrete points in time The model maintains a list of events ("event list") At each step, the scheduled event with the lowest time gets
Time-domain analysis of discrete-time LTI systems Discrete-time signals Di erence equation single-input, single-output systems in discrete time The zero-input response (ZIR): characteristic values and modes The zero (initial) state response (ZSR): the unit-pulse response, convolution System stability The eigenresponse .
