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Lesson_200

# Lesson_200 - Lesson Title Filters Lesson Number 20[Sections...

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Lesson Title: Filters Lesson Number: 20 [Sections 7.7 and 7.10] Background: It has been previously established that a finite impulse response (FIR) filter has an impulse response which persists for only of a finite number of sample values. The impulse response of an N th order FIR is given by: { } 1 1 0 ,..., , ] [ - = N h h h k h 1i which is graphically interpreted in Figure 1 in what is called direct form . The response of an FIR to an arbitrary input time-series x [ k ] is formally given by the linear convolution sum: - = = 1 0 ] - [ ] [ N m m m k x h k y 1ii where the filter coefficients, or tap weight multipliers , are simply the impulse response sample values. Figure 1: N th order FIR filter showing a simulation diagram (left) and impulse response (rigth). Previous studies have shown that the z -transform of a FIR filter satisfies - = - = 1 0 ) ( N k k k z h z H 2i which also leads to a definition of the filter's steady state frequency response given by: ( 29 k j N k k j e z e h e H z H j ω ω ω - - = = = = 1 0 ) ( 2ii which is normally interpreted as magnitude and phase functions over the normalized range ω∈ [- π , π ). Several special cases were studied extensively by the authors and class notes. They are the MA and Comb FIRs.

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Multiplier-less FIRs The simplest FIRs to implement are those which are multiplier-free (i.e., contain only shift- registers and adders). One such FIR is called the moving average (MA) filter and is graphically interpreted in Figure 2. Mathematically a MA filter has a transfer function H ( z ) given by ( 29 1 1 - 1 1 - 1 1 1 1 1 1 1 1 1 1 ) ( 1 1 - 1 - 1 - 0 - 1 0 - - = - = - - - = = - = = - - - - = = - = z z z N z z N z z N z N z N z N z N z H N N N N N m m m m N m m MA 3 Another FIR previously studied was called a comb filter which is also interpreted in Figure 2. A comb filter has a transfer function given by ( 29 N N N C z z z z H 1 1 ) ( ± = ± = - 4 Figure 2: Simple multiplier-free FIR filter architectures showing a moving average (left) and a comb FIR (right). A moving average FIR produces an output which represents the average value of the N most current sample values. The comb filter simply adds or subtracts the value of a delayed sample from that of the current sample. It can be immediately seen that both have a very simple structure and require no multipliers to implement; only shift-registers and adders. In addition, both FIRs have a similar numerator, namely N ( z )=1 ± z - N , where the N roots, or zeros, are given by ( 29 ( 29 ] 1 - , 0 [ for 0 1 ] 1 - , 0 [ for 0 1 / 2 / / 2 N i e z z N i e z z N i j i N N j N i j i N = = - = = + + π π π 5 It can be noted, however, that the MA filter also contains a poles at z =1, which is not present in the comb filters transfer function. The pole-zero distributions of the multiplier-less FIRs are displayed in Figure 3 for the case where N =6. Observe that in all cases the zeros are separated by 2 π /6 radians and that the MA filter has a pole-zero cancellation occur at z =1.
Figure 3.a: 5 th order MA: H(z)=(1-z -7 )/(1-z -1 ) Figure 3.b: 6 th order Comb H(z)=(1-z -7 ) Figure 3.b: 6 th order Comb H(z)=(1+z -7 ) Figure 3: Special FIR cases.

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Lesson_200 - Lesson Title Filters Lesson Number 20[Sections...

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