Signal & Linear system Chapter 3 Time Domain Analysis of DT System Basil Hamed 3.1 Introduction Recall from Ch #1 that a common scenario in todays electronic systems is to do most of the processing of a signal using a computer. A computer cant directly process a C-T signal but instead needs a stream of numberswhich is a D-T signal. Basil Hamed

2 3.1 Introduction What is a discrete-time (D-T) signal? A discrete time signal is a sequence of numbers indexed by integers Example: x[n] n = , -3, -2, -1, 0, 1, 2, 3, Basil Hamed 3 3.1 Introduction

D-T systems allow us to process information in much more amazing ways than C-T systems! sampling is how we typically get D-T signals In this case the D-T signal y[n] is related to the C-T signal y(t) by: T is sampling interval Basil Hamed 4 3.1 Introduction Discrete-time signal is basically a sequence of numbers. They may also arise as a result of

sampling CT time signals. Systems whose inputs and outputs are DT signals are called digital system. x[n], ninteger, time varies discretely Examples of DT signals in nature: DNA base sequence Population of the nth generation of certain species Basil Hamed 5 3.1 Introduction

A function, e.g. sin(t) in continuous-time or sin(2 p n / 10) in discrete-time, useful in analysis A sequence of numbers, e.g. {1,2,3,2,1} which is a sampled triangle function, useful in simulation A piecewise representation, e.g. Basil Hamed 6 Size of a discrete-time signal Power and Energy of Signals Energy signals: all x S with finite energy, i.e.

Power signals: all x S with finite power, i.e. Basil Hamed 7 3.2 Useful Signal Operations Three possible time transformations: Time Shifting Time Scaling Time Reversal

Basil Hamed 8 3.2 Useful Signal Operations Time Shift Delay or shift by integer k: Definition: y[n] = x[n - k] Interpretation:

k 0 graph of x[n] shifted by k units to the right k < 0 graph of x[n] shifted by k units to the left Basil Hamed 9 3.2 Useful Signal Operations Time Shift Signal x[n 1] represents instant shifted version of x[n] Find f[k-5]

Basil Hamed 10 3.2 Useful Signal Operations Time- Reversal (Flip) Graphical interpretation: mirror image about origin Basil Hamed 11 3.2 Useful Signal Operations

Time- Reversal (Flip) Signal x[-n] represents flip version of x[n] Find f[-k] Basil Hamed 12 3.2 Useful Signal Operations Time-scale Find f[2k], f[k/2] Basil Hamed

13 3.3 Some Useful Discrete-time Signal Models Combined Operations 2 Ex; Find a) x[2-n] b) x[3n-4] Solution a) x[2 n] { 1, 3, 2,2,1, 3}

b) Basil Hamed 14 3.3 Some Useful Discrete-time Signal Models Much of what we learned about C-T signals carries over to D-T signals Discrete-Time Impulse Function [n] d[n] n

Basil Hamed 15 3.3 Some Useful Discrete-time Signal Models Discrete-Time Unit Step Function u[n] u[n-k]= Basil Hamed 16

3.3 Some Useful Discrete-time Signal Models Discrete-Time Unit ramp Function r[n] n ,n 0 r[n]= 0 ,n 0 Basil Hamed 17 3.3 Some Useful Discrete-time Signal Models

D-T Sinusoids X[n]=Acos ( n+ ) n+ )) Use upper case omega for frequency of D-T sinusoids What is the unit for ? n + must be in radians n in radiansn + must be in radians n in radiansn + must be in radians n in radiansn in radians n + must be in radians n in radians is how many radians jump for each sample is in radians/sample Basil Hamed 18

3.4 Classification of DT Systems o o o o o Linear Systems Time-invariance Systems Causal Systems Memory Systems Stable Systems Linear Systems:

A ( n+ )DT) system is linear if it has the superposition property: If x1[n] y1[n] and x2[n] y2[n] then ax1[n] + bx2[n] ay1[n] + by2[n] Example: Are the following system linear? y[n]=nx[n] Basil Hamed 19 3.4 Classification of DT Systems Basil Hamed

20 3.4 Classification of DT Systems Time-Invariance A system is time-invariant if a delay (or a time-shift) in the input signal causes the same amount of delay (or timeshift) in the output signal If x[n] y[n] then x[n -n0] y[n -n0] x[n] = x1[n-n0] y[n] = y1[n-n0] Ex. Check if the following system is time-invariant: y[n]=nx[n] Basil Hamed

21 3.4 Classification of DT Systems System is Time Varying Basil Hamed 22 3.4 Classification of DT Systems Causal System A system is causal if the output does not anticipate future values of the input, i.e., if the output at any time depends only on values

of the input up to that time. A system x[n] y[n] is causal if When x1[n] y1[n] x2[n] y2[n] And x1[n] = x2[n] for all n no Then y1[n] = y2[n] for all n no Causal: y[n] only depends on values x[k] for k n. Ex. Check if the following system is Causal: y[n]=nx[n] System is causal because it does not depend on future

Basil Hamed 23 3.4 Classification of DT Systems Memoryless (or static) Systems: System output y[n] depends only on the input at instant n, i.e. y[n] is a function of x[n]. Memory (or dynamic) Systems: System output y[n] depends on input at past or future of the instant n

Ex. Check if the following systems are with memory : i. y[n]=nx[n] ii. y[n] =1/2(x[n-1]+x[n]) i. Above system is memoryless because instantaneous ii. System is with memory Basil Hamed 24 is

3.5 DT System Equations: Difference Equations: We saw that Differential Equations model C-T systems D-T systems are modeled by Difference Equations. A general Nth order Difference Equations looks like this: The difference between these two index values is the order of the difference eq. Here we have: n( n+ )n N) =N Basil Hamed 25 3.5 DT System Equations: Difference equations can be written in two

forms: The first form uses delay y[n-1], y[n-2], x[n-1], y[n]+a1y[n-1]+..+aNy[n-N]= b0x[n]+.+bNx[n-M] Order is Max(N,M) The 2nd form uses advance y[n+1], y[n+2], x[n+1],. y[n+N]+a1y[n+N-1]+..+aNy[n]= bN-Mx[n+m]+. +bNx[n] Basil Hamed Order is Max(N,M)

26 3.5 DT System Equations: Sometimes differential equations will be presented as unit advances rather than delays y[n+2] 5 y[n+1] + 6 y[n] = 3 x[n+1] + 5 x[n] One can make a substitution that reindexes the equation so that it is in terms of delays Substitute n with n -2 to yield y[n] 5 y[n-1] + 6 y[n-2] = 3 x[n-1] + 5 x[n-2] Basil Hamed

27 3.5 DT System Equations: Solving Difference Equations Although Difference Equations are quite different from Differential Equations, the methods for solving them are remarkably similar. Here well look at a numerical way to solve Difference Equations. This method is called Recursionand it is actually used to implement ( n+ )or build) many D-T systems, which is the main advantage of the recursive method. The disadvantage of the recursive method is that it doesnt provide a so-called closed-form solutionin other words, you dont get an equation that describes the output ( n+ )you get a finiteduration sequence of numbers that shows part of the output).

Basil Hamed 28 3.5 DT System Equations: Solution by Recursion We can re-write any linear, constant-coefficient difference equation in recursive form. Here is the form weve already seen for an Nth order difference: Basil Hamed 29

3.5 DT System Equations: Nowisolating the y[n] term gives the Recursive Form: current Output value to be computed Some past output values, with values already known Basil Hamed current & past input

values already received 30 3.5 DT System Equations: Note: sometimes it is necessary to re-index a difference equation using n+k n to get this formas shown below. Here is a slightly different formbut it is still a difference equation: y[n+2]-1.5y[n +1] +y[n]= 2x[n] If you isolate y[n] here you will get the current output value in terms of future output values ( n+ )Try It!)We dont want that!

Soin general we start with the Most Advanced output samplehere it is y[n+2]and re-index it to get only n ( n+ )of course we also have to re-index everything else in the equation to maintain an equation): Basil Hamed 31 3.5 DT System Equations: So here we need to subtract 2 from each sample argument: y[n]-1.5y[n -1] +y[n-2]= 2x[n-2] Now we can put this into recursive form as before. Ex: Solve this difference equation recursively y[n]-1.5y[n -1] +y[n-2]= 2x[n-2]

For x[n]=u[n] unit step And ICs of: Basil Hamed 32 3.5 DT System Equations: Recursive Form: y[n]=1.5y[n -1] -y[n-2]+ 2x[n-2] Basil Hamed 33

3.5 DT System Equations: Ex 3.9 P. 273 y[n+2]-y[n +1] +0.24y[n]= x[n+2]-2x[n+1] y[-1]=2, y[-2]=1, and causal input x[n]=n Solution y[n]=y[n -1] -0.24y[n-2]+ x[n]-2x[n-1] y[0]=y[-1] -0.24y[-2]+ x[0]-2x[-1]= 2-0.24= 1.76 y[1]=y[0] -0.24y[-1]+ x[1]-2x[0]= 1.76 0.24( n+ )2)+ 1- 0= 2.28 : : Basil Hamed

34 Convolution Our Interest: Finding the output of LTI systems (D-T & C-T cases) Our focus in this chapter will be on finding the zero-state solution Basil Hamed 35 3.8 System Response to External Input: (Zero State Response) Convolution:

For discrete case: h[n] = H[d[t]] y[n]= x[n]* h[n]= h[n]* x[n] Notice that this is not multiplication of x[n] and h[n]. Visualizing meaning of convolution: Flip h[k] By shifting h[k] for all possible values of n, pass it through x[n]. Basil Hamed

36 3.8 System Response to External Input: (Zero State Response) For a LTI D-T system in zero state we no longer need the difference equation model-Instead we need the impulse response h[n] & convolution Difference Equation Convolution & Impulse resp

Equivalent Basil Models (for zero state) Hamed 37 3.8 System Response to External Input: (Zero State Response) Properties of DT Convolution: Same as CT Convolution Ex: 3.13 P.289 Find y[n]

Solution h [ ]= U[k]u[n-k]=1 =0 0

From Section B7-4 P49 OR Geometric Sum Basil Hamed 39 3.8 System Response to External Input: (Zero State Response) Graphical procedure for the convolution: Step 1: Write both as functions of k: x[k] & h[k] Step 2: Flip h[k] to get h[-k] Step 3: For each output index n value of interest, shift by n to

get h[n -k] ( n+ )Note: positive n gives right shift!!!!) Step 4: Form product x[k]h[nk] and sum its elements to get the number y[n] Basil Hamed 40 3.8 System Response to External Input: (Zero State Response) Example of Graphical Convolution Find y[n]=x[n]*h[n] for all integer values of n

So..what we know so far is that: y[n] starts at 0 ends at 6 Basil Hamed 41 3.8 System Response to External Input: (Zero State Response) Solution For this problem I choose to flip x[n] My personal preference is to flip the shorter signal although I sometimes dont follow that ruleonly through lots of

practice can you learn how to best choose which one to flip. Step 1: Write both as functions of k: x[k] & h[k] Basil Hamed 42 3.8 System Response to External Input: (Zero State Response) Step 2: Flip x[k] to get x[-k] Commutativity says we can flip either x[k] or h[k] and get the same

answer Here I flipped x[k] Basil Hamed 43 3.8 System Response to External Input: (Zero State Response) We want a solution for n = -2, -1, 0, 1, 2, so must do Steps 3&4 for all n. Butlets first do: Steps 3&4 for n= 0 and then proceed from there. Step 3: For n= 0, shift by n to get x[n-k]

For n= 0 case there is no shift! x[0 -k] = x[-k] Step 4: For n= 0, Form the product x[k]h[nk] and sum its elements to give y[n] Sum over k y[0]=6Basil Hamed 44 3.8 System Response to External

Input: (Zero State Response) Steps 3&4 for n= 1 Step 3: For n= 1, shift by n to get x[n-k] Step 4: For n= 1, Form the product x[k]h[nk] and sum its elements to give y[n] Sum over k y[1]=6+6=12 Basil Hamed 45 3.8 System Response to External

Input: (Zero State Response) Steps 3&4 for n= 2 Step 3: For n= 2, shift by n to get x[n-k] Step 4: For n= 2, Form the product x[k]h[nk] and sum its elements to give y[n] Sum over k y[2]=3+6+6=15 Basil Hamed 46 3.8 System Response to External

Input: (Zero State Response) Steps 3&4 for n= 6 Step 3: For n= 6, shift by n to get x[n-k] Step 4: For n= 6, Form the product x[k]h[nk] and sum its elements to give y[n] Sum over k y[6]=3 Basil Hamed 47 3.8 System Response to External

Input: (Zero State Response) Steps 3&4 for all n > 6 Step 3: For n> 6, shift by n to get x[n-k] Step 4: For n > 6, Form the product x[k]h[nk] and sum its elements to give y[n] Sum over k y[n] = 0 n>6 Basil Hamed 48 3.8 System Response to External

Input: (Zero State Response) Sonow we know the values of y[n] for all values of n We just need to put it all together as a function Here it is easiest to just plot ityou could also list it as a table Basil Hamed 49 3.8 System Response to External Input: (Zero State Response) BIG PICTURE: Sowhat we have just done is found the zero-state output of a system having an impulse response given by this h[n] when the input is given by this x[n]:

Basil Hamed 50 3.8 System Response to External Input: (Zero State Response) EX: given x[n], and h[n], find y[n] Basil Hamed 51 3.8 System Response to External

Input: (Zero State Response) y[n]={1,2,-2,-3,1,1} Basil Hamed 52 3.8 System Response to External Input: (Zero State Response) Exercises : given the following systems Find y[n] i. x[n]={-2,-1,0,1,2}, h[n]={-1,0,1,2}

ii. x[n]={-1,3,-1,-2}, h[n]={-2,2,0,-1,1} Solution: iii. y[n]={2,1,-2,-6,-4,1,4,4} iv. y[n]= x[n]*Basilh[n]={2,-8,8,3,-8,4,1,-2} Hamed 53 3.8-2 Interconnected Systems Example Find h[n] given:

[] Solution: = Basil Hamed 54 3.8-2 Interconnected Systems [] Basil Hamed

55 Comparison of Discrete convolution and Difference Eq. 1. Difference Eq. require less computation than convolution 2. Difference Eq. require less memory 3. Convolutions describe only zero-state responses. (IC=0) Since difference Eq have many advantages over convolutions, we use mainly difference Eq. in studying LTI lumped systems. For distributed system, we have no choice but to

use convolution. Convolution can be used to describe LTI distributed and lumped Basil systems. Where as difference56 Eq Hamed 3.10 System Stability A discrete-time system is BIBO stable if every bounded input sequence x[n] produces a bounded output sequence. LTID with h[n] is BIBO stable if: is finite Ex Find if the following systems are stable;

Basil Hamed 57 3.10 System Stability I. II. = = which is finite if <1, System is stable is not bounded System is BIUBO unstable. III. system is bounded System is stable

Basil Hamed 58 3.10 System Stability Basil Hamed 59