DSP FIRST 2e 5. FIR Filters – Problems with selected Solutions 77


5–1  Unit-step response of FIR system via convolution  Solution  

  • 5–1
 difference equation   FIR   impulse response   filter coefficients 
 

5–2  Length of Convolution  Solution  

  • 5–2
 flip and slide   support 
 

5–3  Impulse response of cascaded LTI systems  Solution  

  • 5–3
 delta   filter coefficients   running average filter 
 

5–4  Difference equation and impulse response of cascaded LTI systems  Solution  

  • 5–4
 delta   FIR 
 

5–5  Block diagram of FIR system and output signal  Solution  

  • 5–5
 impulse response   delta 
 

5–6  Output from FIR Filter for Finite-Length Input Signal ♦ Impulse Response  Solution  

  • 5–6
 impulse   LTI   output 
 

5–7  Output from FIR Filter for Complex Exponential Input Signal  Solution  

  • 5–7
 difference   exponential   output 
 

5–8  Running Average FIR Filter ♦ Step Response  Solution  

  • 5–8
 average   filter   running 
 

5–9  Linearity & Time-Invariance Properties  Solution  

  • 5–9
 frequency   input   LTI   output 
 

5–10  Difference Equation Block Diagram of FIR Filter  Solution  

  • 5–10
 block   difference   filter 
 

5–11  Plot Finite-Length Signal \(x[n]\) Defined by Shifted Impulses  Solution  

  • 5–11
 delay   Dirac   shift   step 
 

5–12  Difference Equation from Block Diagram of FIR Filter  Solution  

  • 5–12
 block   difference   filter 
 

5–13  Linearity & Time-Invariance Used to Construct Output Signal  Solution  

  • 5–13
 input   LTI   output 
 

5–14  Output from FIR Filter for Complex Exponential Input Signal  Solution  

  • 5–14
 exponential   filter   output 
 

5–15  Output of LTI System to \(u[n]\) 

  • 5–15
 FIR difference equation   unit step function   running sum 
 

5–16  Output of LTI System to Complex Exponentials 

  • 5–16
 FIR difference equation   unit step function   running sum 
 

5–17  Output from FIR Filter for Finite-Length Input Signal 

  • 5–17
 difference   FIR filter   impulse response   LTI 
 

5–18  Draw Block Diagrams from FIR Difference Equation 

  • 5–18
 LTI   direct form   transposed direct form 
 

5–19  Determine the Duration of the Output of an FIR Filter 

  • 5–19
 FIR   Filter   Output   support   convolution 
 

5–20  Construct Output via Linearity and Time-Invariance 

  • 5–20
 Linearity   Time-Invariance   FIR filter 
 

5–21  Construct Output via Linearity and Time-Invariance 

  • 5–21
 input   LTI   output   unit step 
 

5–22  Impulse response of cascade of two systems  Solution  

  • 5–22
 cascade   difference equation 
 

5–23  Properties of LTI systems  Solution  

  • 5–23
 linearity   time-invariance   causality 
 

5–24  Output signal using linearity & time-invariance  Solution  

  • 5–24
 FIR   shifted impulse   causality 
 

5–25  Difference equation and \(H(z)\) from impulse response  Solution  

  • 5–25
 convolution   delta   output signal 
 

5–26  Difference equation and \(H(z)\) from impulse response  Solution  

  • 5–26
 convolution   delta   output signal 
 

5–27  Determine the Output of an FIR Filter for a Given Input  Solution  

  • 5–27
 FIR   Filter   Output 
 

5–28  Determine the Duration of the Output of an FIR Filter  Solution  

  • 5–28
 FIR   Filter   Output 
 

5–29  Cascade Connection of LTI Systems  Solution  

  • 5–29
 Cascade   LTI 
 

5–30  Find Outputs of an FIR Filter for Step and Cosine Inputs  Solution  

  • 5–30
 FIR   Filter   Output 
 

5–31  Find Outputs of an FIR Filter for Step and Cosine Inputs  Solution  

  • 5–31
 FIR   Filter   Output 
 

5–32  Find Outputs of an FIR Filter for Step and Cosine Inputs  Solution  

  • 5–32
 FIR   Filter   Output 
 

5–33  Using the Unit Step Sequence to Represent a Finite-Length Signal  Solution  

  • 5–33
 unit step   finite-length 
 

5–34  Determine the Duration of the Output of an FIR Filter  Solution  

  • 5–34
 FIR   filter   impulse response 
 

5–35  Cascade Connection of LTI Systems  Solution  

  • 5–35
 cascade   difference equation   impulse response 
 

5–36  Test for Linearity and Time-Invariance 

  • 5–36
 Linearity   Time-Invariance 
 

5–37  Unit step response of 3-point averager  Solution  

  • 5–37
 finite-length pulse 
 

5–38  Nonzero region of FIR filter output  Solution  

  • 5–38
 convolution   filter coefficients 
 

5–39  Impulse response & difference equation of cascaded systems  Solution  

  • 5–39
 convolution 
 

5–40  FIR filtering of signal defined by its spectrum  Solution  

  • 5–40
 9-point averager   output signal 
 

5–41  Difference equation & block diagram from impulse response  Solution  

  • 5–41
 linear   time-invariant   causal   nonlinear system 
 

5–42  Output from convolution of impulse response & input pulse  Solution  

  • 5–42
 running sum   FIR 
 

5–43  Difference equation & block diagram from impulse response  Solution  

  • 5–43
 linear   time-invariant   causal   nonlinear system 
 

5–44  Output from convolution of impulse response & input pulse  Solution  

  • 5–44
 running sum   FIR 
 

5–45  Difference equation & block diagram from impulse response  Solution  

  • 5–45
 linear   time-invariant   causal   nonlinear system 
 

5–46  Output from convolution of impulse response & input pulse  Solution  

  • 5–46
 running sum   FIR 
 

5–47  Linearity & Time-Invariance Used to Construct Output Signal  Solution  

  • 5–47
 input   LTI   output 
 

5–48  Linearity & Time-Invariance Used to Construct Output Signal  Solution  

  • 5–48
 input   LTI   output 
 

5–49  Output from FIR Filter for Finite-Length Input Signal  Solution  

  • 5–49
 filter   input   LTI   output 
 

5–50  Output from FIR Filter for Complex Exponential Input Signal  Solution  

  • 5–50
 amplitude   difference   exponential   phase 
 

5–51  Output from FIR Filter for Finite-Length Input Signal  Solution  

  • 5–51
 difference   filter   output 
 

5–52  Impulse Response of FIR Filter ♦ Complex Exponential Response  Solution  

  • 5–52
 difference   exponential   filter   impulse 
 

5–53  Output of LTI System to Finite Length Complex Exponential 

  • 5–53
 FIR difference equation 
 

5–54  Output from FIR Filter for Finite-Length Input Signal  Solution  

  • 5–54
 difference equation   convolution   LTI 
 

5–55  Construct Output via Linearity and Time-Invariance  Solution  

  • 5–55
 linearity   time-invariance   FIR filter   causal 
 

5–56  Matching Output Signal to \(h[n]\) or Difference Equation  Solution  

  • 5–56
 impulse response   conv 
 

5–57  Output from FIR Filter for Complex Exponential Input Signal  Solution  

  • 5–57
 difference   exponential   filter   output 
 

5–58  Linearity & Time-Invariance Used to Construct Output Signal  Solution  

  • 5–58
 LTI   ramp   step 
 

5–59  Output from FIR Filter for Complex Exponential Input Signal  Solution  

  • 5–59
 amplitude   difference   exponential   filter   phase 
 

5–60  Running Average FIR Filter ♦ Step Response  Solution  

  • 5–60
 averager   filter   running   step 
 

5–61  Output from FIR Filter for Finite-Length Input Signal  Solution  

  • 5–61
 difference   filter   LTI   output 
 

5–62  Output from FIR Filter for Finite-Length Input Signal  Solution  

  • 5–62
 filter   input   LTI   output 
 

5–63  Output from FIR Filter for Finite-Length Input Signal  Solution  

  • 5–63
 filter   input   LTI   output 
 

5–64  Linearity & Time-Invariance Used to Construct Output Signal  Solution  

  • 5–64
 input   LTI   output 
 

5–65  Output from FIR Filter for Complex Exponential Input Signal  Solution  

  • 5–65
 amplitude   difference   exponential   filter   phase 
 

5–66  Linearity & Time-Invariance Used to Construct Output Signal  Solution  

  • 5–66
 Dirac   LTI   output 
 

5–67  Output from FIR Filter for Finite-Length Input Signal ♦ Impulse Response  Solution  

  • 5–67
 difference   filter   impulse 
 

5–68  Output of LTI System to \(u[n]\)  Solution  

  • 5–68
 FIR difference equation   unit step function   running sum 
 

5–69  Determine the Duration of the Output of an FIR Filter  Solution  

  • 5–69
 FIR   filter   length 
 

5–70  Construct Output via Linearity and Time-Invariance  Solution  

  • 5–70
 linearity   time-invariance   FIR filter   causal 
 

5–71  Find Output of FIR Filter Given Coefficients  Solution  

  • 5–71
 MATLAB   shifted impulse 
 

5–72  Construct Output via Linearity and Time-Invariance  Solution  

  • 5–72
 input   LTI   output   impulse signal 
 

5–73  Output from FIR Filter for Finite-Length Input Signal  Solution  

  • 5–73
 difference equation   convolution   LTI 
 

5–74  Cascade of Two LTI Systems  Solution  

  • 5–74
 blurring filter   impulse response   difference equation 
 

5.1–75  Running Average FIR Filter ♦ Time Response  Solution  

  • 5–75
 averager   running   step 
 

5.6–76  Output from FIR Filter for Finite-Length Input Signal  Solution  

  • 5–76
 filter   impulse   output 
 

5.8–77  Linearity & Time-Invariance Used to Construct Output Signal  Solution  

  • 5–77
 input   LTI   output