Chapter 10

Discrete-Time Linear Time-Invariant Systems

Sections 10.1-10-3

Representation of Discrete-Time Signals• We assume Discrete-Time LTI systems

• The signal X[n] can be represented using unit sample function or unit impulse function: [n]

• Remember:

• Notations:

k

knkxnx ][][][

notes

knkxelseknnx ],[,0][][

1],1[]1[]1[]1[][][

0],0[]0[]0[]0[][][

1

0

nxnxnnxnx

nxnxnnxnx

Representation of Discrete-Time Signals - Example

)]1([]1[][]0[)]1([]1[

][][][][ 101

nxnxnx

nxnxnxnx

Convolution for Discrete-Time Systems• LTI system response can be described using:

• For time-invariant: [n-k]h[n-k]

• For a linear system: x[k][n-k]x[k]h[n-k]

• Remember:

• Thus, for LTI:

• We call this the convolution sum

• Remember:

k

knkxnx ][][][

System[n] h[n]

][*][][][][][][][ nhnxknhkxnyknkxnxkk

][*][][][][

][*][][][][

nxnhknxkhny

nhnxknhkxny

k

k

][][*][][*][ 000 nnhnnnhnnnh

Impulse Response of a System

Convolution for Discrete-Important Properties• By definition

• Remember (due to time-invariance property):

• Multiplication

][][*][][*][ 000 nnhnnnhnnnh

][][*][][ nhnnhny

][][][][ 00 ngnnngn

Properties of Convolution

• Commutative • Associative • Distributive

]0[]0[]1[]1[]2[]2[

....]1[]1[]0[]0[]1[]1[]2[]2[]3[]3[...

][*][][][][

][*][][][][

hnxhnxhnx

hnxhnxhnxhnxhnx

nxnhknxkhny

nhnxknhkxny

k

k

Example• Given the following block diagram

– Find the difference equation

– Find the impulse response: h[n]; plot h[n]

– Is this an FIR (finite impulse response) or IIR system?

– Given x[1]=3, x[2]=4.5, x[3]=6, Plot y[n] vs. n

– Plot y[n] vs. n using Matlab

• Difference equation

• To find h[n] we assume x[n]=[n], thus y[n]=h[n]

– Thus: h[0]=h[1]=h[2]=1/3

• Since h[n] is finite, the system is FIR• In terms of inputs:

])2[]1[][(3

1][ nxnxnxny

Figure 10.3

])2[]1[][(3

1][][ nnnnhny

Figure 10.3

FIR system contains finite number of nonzero terms

Example – cont.• Given the following block diagram

– Find the difference equation

– Find the impulse response: h[n]; plot h[n]

– Is this an FIR (finite impulse response) or IIR system?

– Given x[1]=3, x[2]=4.5, x[3]=6, Plot y[n] vs. n

– Plot y[n] vs. n using Matlab

• In terms of inputs:

• Calculate for n=0, n=1, n=2, n=3, n=4, n=5, n=6– n=0; y[0]=0

– n=1; y[n]=1

– n=2; y[2]=2.5

– n=3; y[2\3]=4.5

– n=4; y[4]=3.5

– n=5; y[5]=2

– n=6; y[6]=0

•

Figure 10.3

]0[]0[]1[]1[]2[]2[

....]1[]1[]0[]0[]1[]1[]2[]2[]3[]3[...][

hnxhnxhnx

hnxhnxhnxhnxhnxny

Try for different values of n

Example – cont. (Graphical Representation)

]0[]0[]1[]1[]2[]2[

....]1[]1[]0[]0[]1[]1[]2[]2[]3[]3[...][

hnxhnxhnx

hnxhnxhnxhnxhnxny

h[0]=h[1]=h[2]=1/3 x[1]=3, x[2]=4.5, x[3]=6

X[n-k]X[m]

Example

• Consider the following difference equation:y[n]=ay[n-1]+x[n]– Draw the block diagram of this system – Find the impulse response: h[n]– Is it a causal system?– Is this an IIR or FIR system?

Example

• Consider the following difference equation:y[n]=ay[n-1]+x[n]; – Draw the block diagram of this system – Find the impulse response: h[n]– Is this an IIR or FIR system?

We assume x[n]=[n]y[n]=h[n]=ah[n-1]+[n];y[0]=h[0]=1y[1]=h[1]=ay[2]=h[2]=a^2y[3]=h[3]= a^3

h[n]=a^n ; n>=0

It is IIR (unbounded)

Causal system (current and past)

Example

• Assume h[n]=0.6^n*u[n] and x[n]=u[n]– Find the expression for y[n]

– Plot y[n]

– Plot y[n] using Matlab

0];6.01[5.26.0][][6.0

][*][][][][

1

0

nkuknu

nxnhknxkhny

nn

k

k

k

k

k y[0]=1y[1]=1.6…..y(100)=2.5 Steady State Value is 2.5

h[n]x[n] y[n]

Remember These Geometric Series:

Properties of Discrete-Time LTI Systems

• Memory: – A memoryless system is a pure gain

system: iff h[n]=K[n]; • K=h[0] = constant & h[n]=0 otherwise

• Causality – y[n] has no dependency on future values

of x[n]; thus h[n]=0 for n<0 (note h[n] is non-zero only for [n=0].

][*][][][][

][*][][][][

nxnhknxkhny

nhnxknhkxny

k

k

...]1[]1[][]0[...]0[][][][][0

nhxnhxhnxknxkhnyk

...]1[]1[][]0[...]0[][][][][

nhxnhxhnxknkkxnyn

k

]0[]0[]1[]1[]2[]2[

....]1[]1[]0[]0[]1[]1[]2[]2[]3[]3[...][

hnxhnxhnx

hnxhnxhnxhnxhnxny

Note that if k<0depending on future; Thus h[k] should be zero to remove dependency on the future.

Properties of Discrete-Time LTI Systems

• Stability – BIBO: |x[n]|< M

– Absolutely summable:

• Invertibility: – If the input can be determined from output

– It has an inverse impulse response

– Invertible if there exists: hi[n]*h[n]=[n]

k

kh |][|

Example 1

• Assume h[n]= u[n] (1/2)^n– Memoryless?

– Casual system?

– Stable?

– Has memory (dynamic): h[n] is not K[n] (not pure gain); h[n] is non-zero

– Is causal: h[n]=0 for n<0

– Stable:

k k

kkh0

25.01

1|2

1||][|

h[n]x[n] y[n]

Example 2

• Assume h[n]= u[n+1] (1/2)^n– Memoryless?

– Casual system?

– Stable?

– Has memory (dynamic): h[n] is not K[n] (not pure gain)

– Is NOT causal: h[n] not 0 for n<0; h[-1]=2

– Stable:

k k

kkh1

45.01

12|

2

1||][|

h[n]x[n] y[n]

Example 3

• Assume h[n]= u[n] (2)^n– Memoryless?

– Casual system?

– Stable?

– Has memory (dynamic): h[n] is not K[n] (not pure gain)

– Is causal: h[n]=0 for n<0

– Not Stable:

k k

kkh0

|2||][|

h[n]x[n] y[n]