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Chapter 10

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 : d [n] Remember: Notations: . notes.

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Chapter 10

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  1. Chapter 10 Discrete-Time Linear Time-Invariant Systems Sections 10.1-10-3

  2. Representation of Discrete-Time Signals • We assume Discrete-Time LTI systems • The signal X[n] can be represented using unit sample function or unit impulsefunction: d[n] • Remember: • Notations: notes

  3. Representation of Discrete-Time Signals - Example

  4. Convolution for Discrete-Time Systems • LTI system response can be described using: • For time-invariant: d[n-k]h[n-k] • For a linear system: x[k]d[n-k]x[k]h[n-k] • Remember: • Thus, for LTI: • We call this the convolution sum • Remember: System d[n] h[n] Impulse Response of a System

  5. Convolution for Discrete-Important Properties • By definition • Remember (due to time-invariance property): • Multiplication

  6. Properties of Convolution • Commutative • Associative • Distributive

  7. 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]=d[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: Figure 10.3 Figure 10.3 FIR system contains finite number of nonzero terms

  8. 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 Try for different values of n

  9. Example – cont. (Graphical Representation) X[m] X[n-k] h[0]=h[1]=h[2]=1/3 x[1]=3, x[2]=4.5, x[3]=6

  10. 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?

  11. 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]=d[n] y[n]=h[n]=ah[n-1]+d[n]; y[0]=h[0]=1 y[1]=h[1]=a y[2]=h[2]=a^2 y[3]=h[3]= a^3 h[n]=a^n ; n>=0 It is IIR (unbounded) Causal system (current and past)

  12. 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 h[n] y[n] x[n] y[0]=1 y[1]=1.6 ….. y(100)=2.5  Steady State Value is 2.5

  13. Remember These Geometric Series:

  14. Properties of Discrete-Time LTI Systems • Memory: • A memoryless system is a pure gain system: iff h[n]=Kd[n]; • K=h[0] = constant & h[n]=0 otherwise • Causality • y[n] has no dependency on future values of x[n]; thush[n]=0 for n<0 (note h[n] is non-zero only for d[n=0]. Note that if k<0depending on future; Thus h[k] should be zero to remove dependency on the future.

  15. 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]=d[n]

  16. Example 1 • Assume h[n]= u[n] (1/2)^n • Memoryless? • Casual system? • Stable? • Has memory (dynamic): h[n] is not Kd[n] (not pure gain); h[n] is non-zero • Is causal: h[n]=0 for n<0 • Stable: h[n] y[n] x[n]

  17. Example 2 • Assume h[n]= u[n+1] (1/2)^n • Memoryless? • Casual system? • Stable? • Has memory (dynamic): h[n] is not Kd[n] (not pure gain) • Is NOT causal: h[n] not 0 for n<0; h[-1]=2 • Stable: h[n] y[n] x[n]

  18. Example 3 • Assume h[n]= u[n] (2)^n • Memoryless? • Casual system? • Stable? • Has memory (dynamic): h[n] is not Kd[n] (not pure gain) • Is causal: h[n]=0 for n<0 • Not Stable: h[n] y[n] x[n]

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