Simple Digital Filters Chapter 7B Part B Simple FIR Digital Filters LTI Discrete-Time Systems in the Transform-Domain Simple Digital Filters Simple IIR Digital Filters Comb Filters 3. Simple FIR Digital Filters. Simple FIR Digital Filters. Lowpass FIR Digital Filters Later in the course we shall review various methods of designing frequency-selective filters satisfying prescribed specifications We now describe several low-order order FIR and IIR digital filters with reasonable selective frequency responses that often are satisfactory in a number of applications 4 FIR digital filters considered here have integer-valued impulse response coefficients (quantified) These filters are employed in a number of practical applications, primarily because of their simplicity, which makes them amenable to inexpensive hardware implementations 5 The simplest lowpass FIR digital filter is the -point moving-average filter given by H ( ) The above transfer function has a ero at = and a pole at = Note that here the pole vector has a unity magnitude for all values of, thus ( ).5 H e e 6
. Lowpass FIR Digital Filters. Lowpass FIR Digital Filters. Lowpass FIR Digital Filters Im Re - - As increases from to, the magnitude of the ero vector decreases from a value of, the diameter of the unit circle, to We can work out the frequency response / H ( e ) e cos( /) 7 / c passband log / 3dB c stopband 3-dB cutoff frequency normalied digital angular frequency 8 / A cascade of 3 sections an improved scheme c Notice: The cascade of firstorder sections yields a sharper magnitude response but at the expense of a decrease in the width of the passband 9. Lowpass FIR Digital Filters M-order FIR Lowpass (M-order moving-average) Filter / M H M m m h( n) R ( n) M M. Highpass FIR Digital Filters The simplest highpass FIR filter is obtained from the simplest lowpass FIR filter by replacing with This results in H ( ) Corresponding frequency response is given by H e / ( ) e sin( /). Highpass FIR Digital Filters Improved highpass magnitude response can again be obtained by cascading several sections of the first-order highpass filter Alternately, a higher-order highpass filter of the form M n n H M n is obtained by replacing with in the transfer function of a moving average filter
. Highpass FIR Digital Filters. Simple IIR Digital Filters. Lowpass IIR Digital Filters / Lowpass IIR Digital Filters Highpass IIR Digital Filters Bandpass IIR Digital Filters Bandstop IIR Digital Filters Higher-order IIR Digital Filters A first-order causal lowpass IIR digital filter has a transfer function given by HLP where a < for stability The above transfer function has a ero at = i.e., at which is in the stopband H ( ) has a real pole at LP 3 4 5. Lowpass IIR Digital Filters. Lowpass IIR Digital Filters. Highpass IIR Digital Filters As w increases from to p, the magnitude of the ero vector decreases from a value of to, whereas, for a positive value of a, the magnitude of the pole vector increases from a value of to The maximum value of the magnitude function is at w =, and the minimum value is at w = p.8.6.4. First-order IIR Lowpass Filter =.8 =.7 =.5..4.6.8 / Gain, db -5 - First-order IIR Lowpass Filter -5 =.5 =.7 =.8 - - - / A first-order causal highpass IIR digital filter has a transfer function given by HHP where a < for stability The above transfer function has a ero at = i.e., at w = which is in the stopband 6 7 8
. Highpass IIR Digital Filters and gain responses of H ( LP ) are shown below First-order IIR Highpass Filter.8.6.4 =.8. =.7 =.5..4.6.8 / Gain,dB First-order IIR Highpass Filter -5 - -5 =.8 =.7 =.5 - - - / 9.3 Bandpass IIR Digital Filters A nd-order bandpass digital transfer function is given by HBP ( ) Its squared magnitude function is HBP ( e ) ( ) ( cos ) [ ( ) ( ) cos cos ].3 Bandpass IIR Digital Filters H ( goes to ero at and BP e ) It assumes a maximum value of at, called the center frequency of the bandpass filter, where The frequencies and c c where the squared magnitude becomes / are called the 3-dB cutoff frequencies The difference between the two cutoff frequencies, is called the 3-dB bandwidth.3 Bandpass IIR Digital Filters.4 Bandstop IIR Digital Filters.4 Bandstop IIR Digital Filters The transfer function is a BR function if and.8.6.4. Second-order IIR Bandpass Filter (=.34) =. =.5 =.8..4.6.8 /.8.6.4. Second-order IIR Bandpass Filter (=.6) =. =.5 =.8..4.6.8 / A nd-order bandstop digital filter has a transfer function given by HBS ( ) The transfer function is a BR function if and Its magnitude response is plotted in the next slide 3 Here, the magnitude function takes the maximum value of at w = and w = p It goes to at, where, called the notch frequency, is given by cos The digital transfer function is more commonly called a notch filter The difference between the two cutoff frequencies is called the 3-dB notch bandwidth 4
.4 Bandstop IIR Digital Filters.5 Higher-Order IIR Digital Filters.5 Higher-Order IIR Digital Filters Second-order IIR Bandstop Filter (=.5).8.6.4 =.. =.5 =.8..4.6.8 / Second-order IIR Bandstop Filter (=.5).8.6.4 =.. =.5 =.8..4.6.8 / 5 By cascading the simple digital filters discussed so far, we can implement digital filters with sharper magnitude responses Consider a cascade of K first-order lowpass sections characteried by the transfer function K GLP 6 The corresponding squared-magnitude function is given by K ( ) (cos ) GLP ( e ) ( cos ) To determine the relation between its 3-dB cutoff frequency and the parameter,we set c K ( ) (cos c ) c ( cos ) 7.5 Higher-Order IIR Digital Filters which when solved for a, yields for a stable G ( LP ) : ( C)cosc sinc CC ) C cos where C ( )/ K K It should be noted that the expression for a given earlier reduces to sinc for K= cos c c 8 The simple filters discussed so far are characteried either by a single passband and/or a single stopband There are applications where filters with multiple passbands and stopbands are required The comb filter is an example of such filters 9 In its most general form, a comb filter has a frequency response that is a periodic function of w with a period p/l, where L is a positive integer If H() is a filter with a single passband and/or a single stopband, a comb filter can be easily generated from it by replacing each delay in its realiation with L delays resulting in a structure with a transfer function given by G()=H( L ) 3
If He ( ) exhibits a peak at p, then Ge ( ) will exhibit L peaks at k / L, p k L in the frequency range Likewise, if He ( ) has a notch at,then He ( ) will have L notches at k/ L, k Lin the frequency range A comb filter can be generated from either an FIR or an IIR prototype filter 3 For example, the comb filter generated from the prototype lowpass FIR filter H (/)( ) has a transfer function G ( ) (/)( ) L ( G has L notches at e ) (k) / Land L peaks at k / L, k L, in the frequency range.8.6.4. Comb Filter from Lowpass Prototype (L=5).5.5 / 3 For example, the comb filter generated from the prototype highpass FIR filter H (/)( ) has a transfer function G ( ) (/)( ) L ( G has L peaks at e ) (k ) / L and L notches at k / L k L, in the frequency range.8.6.4. Comb Filter from Highpass Prototype (L=5).5.5 33 / Depending on applications, comb filters with other types of periodic magnitude responses can be easily generated by appropriately choosing the prototype filter For example, the M-point moving average filter M H M( ) has been used as a prototype 34 This filter has a peak magnitude at w =, and M notches at l/ M, l M The corresponding comb filter has a transfer function LM G ( ) L M( ) whose magnitude has L peaks at k / L, k Land L(M ) notches at k / LM k L( M ) 35 Comb Filter from M-point moving avarage Prototype.8.6.4. L=5 M=3.5.5 / 36