Dr-Ing Sudchai Boonto Department of Control System and Instrumentation Engineering King Mongut s Unniversity of Technology Thonburi Thailand
Outline Introduction Some Useful Discrete-Time Signal Models Size of a Discrete-Time Signal Useful Signal Operations Examples 2/36
Introduction Signals defined only at discrete instants of time are discrete-time signals We consider uniformly spaced discrete instants such as, 2T, T,, T, 2T, 3T,, T, Discrete-time signals can be specified as f(t ), y(t ), where are integer Frequently used notation are f[], y[], etc, where they are understood that f[] = f(t ), y[] = y(t ) and that are integers Typical discrete-time signals are just sequences of numbers a discrete-time system may seen as processing a sequence of numbers f[] and yielding as output another sequence of numbers y[] 3/36
Discrete-time signal Example a continuous-time exponential f(t) = e t, when sampled every T = second, results in a discrete-time signal f(t ) given by f(t ) = e T = e This is a function of and may be expressed as f[] f[] or f(t ) = e T 2 4 6 8 2T T 4T 6T 8T t 4/36
Discrete-time signal Example Discrete-time signals arise naturally in situations which are inherently discrete-time such as population studies, amortization problems, national income models, and radar tracing They may also arise as a result of sampling continuous-time signals in sampled data systems, digital filtering, etc 5/36
Some Useful Discrete-time signal Models Discrete-Time Impulse Function δ[] The discrete-time counterpart of the continuous-time impulse function δ(t) is δ[], defined by, = δ[] =, δ[] δ[ m] m Unlie its continuous-time counterpart δ(t), this is a very simple function without any mystery 6/36
Some Useful Discrete-time signal Models Discrete-Time Unit Function u[] The discrete-time counterpart of the unit step function u(t) is u[], defined by u[] = {,, < u[] 2 2 3 4 5 6 7 8 If we want a signal to start at =, we need only multiply the signal with u[] 7/36
Some Useful Discrete-time signal Models Discrete-Time Exponential γ a continuous-time exponential e λt can be expressed in an alternate form as For example, e λt = γ t (γ = e λ or λ = ln γ) e 3t = (e 3 ) t = (748) t 4 t = e 386t because ln 4 = 386 that is e 386 = 4 In the study of continuous-time signals and systems we prefer the form e λt rather that γ t 8/36
Some Useful Discrete-time signal Models Discrete-Time Exponential γ cont The discrete-time exponential can also be expressed in two form as e λ = γ (γ = e λ or λ = ln γ) For example e 3 = (e 3 ) = (286) 5 = (e 69 ) because 5 = e 69 In the study of discrete-time signals and systems, the form γ proves more convenient than the form e λ 9/36
Some Useful Discrete-time signal Models Discrete-Time Exponential γ cont If γ =, then = γ = γ = γ = = If γ <, then the signal decays exponentially with 2 8 8 6 4 (8) 6 4 ( 8) 2 2 4 2 6 8 2 4 6 8 2 4 6 8 /36
Some Useful Discrete-time signal Models Discrete-Time Exponential γ cont If γ >, then the signal grows exponentially with 3 3 25 2 2 () 5 ( ) 5 2 2 4 6 8 3 2 4 6 8 /36
Some Useful Discrete-time signal Models Discrete-Time Exponential γ cont the exponential (5) decays faster than (8) ( ) γ = for example the exponential (5) can be expressed γ as 2 (8) 2 8 6 4 2 ( 8) 5 5 2 4 6 8 2 4 6 8 (5) 8 6 4 2 ( 5) 5 5 2 4 6 8 2 4 6 8 2/36
Some Useful Discrete-time signal Models Discrete-Time Sinusoid cos(ω + θ) A general discrete-time sinusoid can be expressed as C cos(ω + θ), where C is the amplitude, Ω is the frequency (in radians per sample), and θ is the phase (in radians) 8 6 ) 4 cos ( π 2 + π 4 2 2 4 6 8 3 2 2 3 3/36
Some Useful Discrete-time signal Models Discrete-Time Sinusoid cos(ω + θ) cont Because cos( x) = cos(x) cos( Ω + θ) = cos(ω θ) It shows that both cos(ω + θ) and cos( Ω + θ) have the same frequency (Ω) Therefor, the frequency of cos(ω + θ) is Ω Sampled continuous-time sinusoid yields a discrete-time sinusoid f[] = cos ωt = cos Ω where Ω = ωt 4/36
Some Useful Discrete-time signal Models Exponentially varying discrete-time sinusoid γ cos(ω + θ) γ cos(ω + θ) is a sinusoid cos(ω + θ) with an exponentially varying amplitude γ ) ) (9) cos ( π 6 π 3 () cos ( π 6 π 3 6 4 2 2 4 6 6 4 2 2 4 6 5 5 5 5 5 5 5 5 5/36
Size of a Discrete-Time Signal Energy signal The size of a discrete-time signal f[] can be measured by its energy E f defined by E f = = f[] 2 the measure is meaningful if the energy of a signal is finite A necessary condition for the energy to be finite is that the signal amplitude must approach as Otherwise the sum will not converge If E f is finite, the signal is called an energy signal 6/36
Size of a Discrete-Time Signal Power signal For the cases, the amplitude of f[] does not approach to as, then the signal energy is infinite, and a measure of the signal will be the time average of the energy (if it exists) the signal power P f is defined by P f = lim N 2N + N = N f[] 2 For periodic signals, the time averaging need be performed only over one period in view of the periodic repetition of the signal If P f is finite and nonzero, called a power signal A discrete-time signal can either be an energy signal or a power signal Some signals are neither energy nor power signals 7/36
Size of a Discrete-Time Signal Example Show that the signal a u[] is an energy signal of energy a 2 if a < It is a power signal of power P f = 5 if a = It is neither an energy signal nor a power signal if a > Solution: E f = = f[] 2 = a 2 = If a < then a approaches to and S = a 2 = + a 2 + a 4 + a 6 + = a 2 S = a 2 + a 4 + a 6 + Subtracting both equations, we have ( a 2 )S = S = a 2 8/36
Size of a Discrete-Time Signal Example cont If a =, then the summation approaches to and N P f = lim f[] 2 N 2N + = N N = lim N 2N + = 2 = 5 = a 2 = lim N N + 2N + If a > lim N 2N + N a 2 = = 9/36
Useful Signal Operations Time Shifting To time shift a signal f[] by m units, we replace with m Thus, f[ m] represents f[] time shifted by m units If m is positive, the shift is to the right (delay) If m is negative, the shift is to the left (advance) Thus f[ 5] is f[] delayed by 5 units The signal is the same as f[] with replaced by 5 Now, f[] = (9) for 3 Therefore f d [] = (9) 5 for 3 5 or 8 5 Thus f[ + 2] is f[] advanced by 2 units The signal is the same as f[] with replaced by + 2 Now, f[] = (9) for 3 Therefore f a [] = (9) +2 for 3 + 2 or 8 2/36
Useful Signal Operations Time Shifting cont f[] 5 (9) 5 5 5 fd[] = f[ 5] fa[] =f[+2] 5 5 5 5 5 5 5 5 (9) +2 (9) 5 2/36
Useful Signal Operations Time Inversion (or Reversal) To time invert a signal f[], we replace with This operation rotates the signal about the vertical axis If f r [] is a time-inverted signal f[], then the expression of f r [] is the same as that for f[] with replaced by Because f[] = (9) for 3, f r [] = (9) for 3 ; that is 3, as shown in Figure on the next slide 22/36
Useful Signal Operations Time Inversion (or Reversal) cont 2 8 f[] 6 4 2 5 5 5 5 fr[] = f[ ] 8 6 4 2 5 5 5 5 23/36
Useful Signal Operations Time Scaling Unlie the continuous-time signal, the discrete-time argument can tae only integer values Some changes in the procedure are necessary Time Compression: Decimation or Downsampling Consider a signal f c [] = f[2] the signal f c [] is the signal f[] compressed by a factor 2 Observe that f c [] = f[], f c [] = f[2], f c [2] = f[4], and so on This fact shows that f c [] is made up of even numbered samples of f[] The odd numbered samples of f[] are missing 24/36
Useful Signal Operations Time Compression: Decimation or Downsampling This operation loses part of the data, and the time compression is called decimation or downsampling In general, f[m] (m integer) consists of only every mth sample of f[] 6 6 5 5 f[] 4 3 2 fc[] = f[2] 4 3 2 5 5 2 5 5 2 25/36
Useful Signal Operations Time Expansion Consider a signal f e [] = f [ ] 2 the signal f e [] is the signal f[] expanded by a factor 2 f e [] = f[], f e [] = f[/2], f e [2] = f[], f e [3] = f[3/2], f e [4] = f[2], f e [5] = f[5/2], f e [6] = f[3], and so on Since, f[] is defined only for integer values of, and is zero (or undefined) for all fractional values of Therefor for the odd numbered samples f e [], f e [3], f e [5], are all zero In general, a function f e [] = f[/m] is defined for =, ±m, ±2m, ±3m, and is zero for all remaining values of 26/36
Useful Signal Operations Time Expansion 6 4 f[] 2 5 5 2 25 3 35 4 6 2 fe[] =f [ ] 4 2 5 5 2 25 3 35 4 27/36
Useful Signal Operations Interpolation The missing samples of the discrete-time signal can be reconstructed from the nonzero valued samples using some suitable interpolations formula In practice, we may use a linear interpolation For example f i [] is taen as the mean of f i [] and f i [2] f i [3] is taen as the mean of f i [2] and f i [4], and so on The process of time expansion and inserting the missing samples using an interpolation is called interpolation or upsampling In this operation, we increase the number of samples 28/36
Useful Signal Operations Time Expansion ] 2 fe[] =f [ fi[] 6 5 4 3 2 5 5 2 25 3 35 4 6 5 4 3 2 5 5 2 25 3 35 4 Interpolation (Upsampling) 29/36
Examples of Discrete-Time Systems Example: money deposit A person maes a deposit (the input) in a ban regularly at an interval of T (say month) The ban pays a certain interest on the account balance during the period T and mails out a periodic statement of the account balance (the output) to the depositor Find the equation relating the output y[] (the balance) to the input f[] (the deposit) In this case, the signals are inherently discrete-time Let f[] = the deposit made at the th discrete instant y[] = the account balance at the th instant computed immediately after the th deposit f[] is received r = interest per dollar per period T The balance y[] is the sum of (i) the previous balance y[ ], (ii) the interest on y[ ] during the period T, and (iii) the deposit f[] y[] = y[ ] + ry[ ] + f[] = ( + r)y[ ] + f[] 3/36
Examples of Discrete-Time Systems Example: money deposit cont or y[] ay[ ] = f[] a = + r another form Bloc diagram or hardware realization y[ + ] ay[] = f[ + ] f[] y[] ay[ ] a y[ ] z Assume y[] is available Delaying it by one sample, we generate y[ ] We generate y[] from f[] and y[ ] 3/36
Examples of Discrete-Time Systems Example: number of students enroll in a course In the th semester, f[] number of students enroll in a course requiring a certain textboo The publisher sells y[] new copies of the boo in the th semester On the average, one quarter of students with boo in saleable condition resell their boos at the end of semester, and the boo life is three semesters Write the equation relating y[], the new boos sold by the publisher, to f[], the number of students enrolled in the th semester, assuming that every student buys a boo In the th semester, the total boos f[] sold to students must be equal to y[] (new boos form the publisher) plus used boos from students enrooled in the two previous semesters There are y[ ] new boo sold in the ( )st semester, and one quarter of these boos; that is, y[ ] will be resold in the th semester 4 Also y[ 2] new boos are sold in the ( 2)nd semester, and one quarter of these; that is y[ 2] will be resold in the ( )st semester 4 Again a quarter of these; that is y[ 2] will be resold in the th semester 6 Therefore, f[] must be equal to the sum of y[], 4 y[ ], and y[ 2] 6 32/36
Examples of Discrete-Time Systems Example: number of students enroll in a course cont y[] + 4 y[ ] + y[ 2] = f[] 6 Above equation can be rewritten as y[ + 2] + y[ + ] + y[] = f[ + 2] 4 6 To mae a bloc diagram the equation is rewritten as y[] = 4 y[ ] y[ 2] + f[] 6 f[] y[] y[ ] y[ 2] z z 4 6 33/36
Examples of Discrete-Time Systems Example: Discrete-Time Differentiator Design a discrete-time system to differentiate continuous-time signals Since Therefore, at t = T y(t) = df dt y(t ) = df dt = lim [f(t ) f[( )T ]] t=t T T By fixing the interval T, the above equation can be expressed as y[] = lim {f[] f[ ]} T T In practice, the sampling interval T cannot be zero Assuming T to be sufficiently small, the above equation can be expressed as y[] {f[] f[ ]} T 34/36
Examples of Discrete-Time Systems Example: Discrete-Time Differentiator cont f(t) f[] y[] C/D D/C T f[ ] y(t) z Discrete-Time Differentiator Bloc Diagram 35/36
Reference Lathi, B P, Signal Processing & Linear Systems, Bereley-Cambridge Press, 998 36/36