LINEAR PHASE FILTERS 1.1 2008/06/10 21:46:40.390 GMT-5 2008/06/26 01:17:22.890 GMT-5 Phuong Huu Nguyen nhphuong@hcmuns.edu.vn Phuong Huu Nguyen nhphuong@hcmuns.edu.vn The ability to have a guaranteed linear phase response is an important advantage of FIR filters over IIR ones. This section presents this matter before we go into FIR filter design in subsequent sections. Let’s denote by ∣H(ω)∣ size 12{ lline H \( ω \) rline } {} the magnitude response and by Φ(ω) size 12{Φ \( ω \) } {} or ∠ size 12{∠} {}H(ω) size 12{H \( ω \) } {}the phase response of a filter.

Phase delay It’s well known to us that in the time domain the output signal is given by the time convolution y ( n ) = x ( n ) ∗ h ( n ) = h ( n ) ∗ x ( n ) size 12{y \( n \) =x \( n \) * h \( n \) =h \( n \) * x \( n \) } {} which is transformed into the frequency domain as Y(ω)=H(ω)X(ω) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMfacaGGOaGaeqyYdCNaaiykaiabg2da9iaadIeacaGGOaGaeqyYdCNaaiykaiaadIfacaGGOaGaeqyYdCNaaiykaaaa@42E2@ The magnitude and phase of the output signal are, respectively, ∣Y(ω)∣=∣H(ω)∣∣X(ω)∣ size 12{ lline Y \( ω \) rline = lline H \( ω \) rline lline X \( ω \) rline } {} ∠ Y ( ω ) =∠ H ( ω ) +∠ X ( ω ) size 12{∠Y \( ω \) "=∠"H \( ω \) "+∠"X \( ω \) } {} By the above phase relation , the phase ∠H(ω) size 12{∠H \( ω \) } {} has the meaning of a phase shift (delay or advance). Because this phase shift depends on the frequency, different frequency components of the input signal may suffer different phase shifts when the signal goes throught the filter , resulting in a distorted output waveform. This waveshape distortion is called phase distortion. Consider an analog sinusoid having period T 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaaaaa@37A1@ sec, and angular frequency ω0 size 12{ω rSub { size 8{0} } } {} rad/sec, ( ω 0 =2π/ T 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacIcacqaHjpWDdaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIYaGaeqiWdaNaai4laiaadsfadaWgaaWcbaGaaGimaaqabaGccaGGPaaaaa@3FF3@ . Since a period corresponds to a phase change of 2π size 12{2π} {} rad, a phase delay of say Φ0 size 12{Φ rSub { size 8{0} } } {} will correspond to a time delay of τp=(Φ0/2π)T0 size 12{τ rSub { size 8{p} } = \( Φ rSub { size 8{0} } /2π \) T rSub { size 8{0} } } {} sec. Thus a phase delay can be interpreted as a time delay. The phase delay of a filter is defined as the negative of the phase Φ(ω) size 12{Φ \( ω \) } {} divided by the corresponding frequency ω size 12{ω} {}: τ p (ω) =  - Φ(ω) ω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabmGadiaacaqabeaadaqaaqaaaOqaaiabes8a0naaBaaaleaacaWGWbaabeaakiaacIcacqaHjpWDcaGGPaWaaCbiaeaacqGH9aqpaSqabeaaaaGccaaMe8UaaiylamaalaaabaGaeuOPdyKaaiikaiabeM8a3jaacMcaaeaacqaHjpWDaaaaaa@4611@ Notice that even called phase delay but τp(ω) size 12{τ rSub { size 8{p} } \( ω \) } {} is really a time delay. A filter is said to have linear phase (or linear phase response) when its phase frequency response is proportional to the negative of the frequency: Φ(ω)=∠H(ω)=−αω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabfA6agjaacIcacqaHjpWDcaGGPaGaeyypa0JaeyiiIaTaamisaiaacIcacqaHjpWDcaGGPaGaeyypa0JaeyOeI0IaeqySdeMaeqyYdChaaa@467C@ This means H(ω)= H 0 e −jαω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIeacaGGOaGaeqyYdCNaaiykaiabg2da9iaadIeadaWgaaWcbaGaaGimaaqabaGccaWGLbWaaWbaaSqabeaacqGHsislcaWGQbGaeqySdeMaeqyYdChaaaaa@42FB@ where α size 12{α} {} is a constant and H 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIeadaWgaaWcbaGaaGimaaqabaaaaa@379A@ is a gain factor independent of frequency . If α size 12{α} {} is possitive the filter delays the input signal, if negative it advances the input signal. The phase delay is now τ p (ω)=− Φ(ω) ω =α MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeGabiGaciaacaqaaeaadaqaaqaaaOqaaiabes8a0naaBaaaleaacaWGWbaabeaakiaacIcacqaHjpWDcaGGPaGaeyypa0JaeyOeI0YaaSaaaeaacqqHMoGrcaGGOaGaeqyYdCNaaiykaaqaaiabeM8a3baacqGH9aqpcqaHXoqyaaa@470F@ Thus phase delay is constant, meaning that all frequency components of the input signal suffer the same time delay, thus output frequency components will be assembled into the same waveform as that of the input (we can check this by , for example, considering an input signal composed a fundamental simusoid and a third harmonic). Thus the output signal is the delayed version of the input . In a filter with a nonlinear phase characterstic, the output waveform will be distorted compared to the input . Of course since a purely simuoidal signal has only one frequency component the output waveform will be exactly the same regardless the filter phase characteristic. The following case is also considered as linear phase Φ(ω) = -αω+β MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabfA6agjaacIcacqaHjpWDcaGGPaGaaGjbVlabg2da9iaaysW7caGGTaGaeqySdeMaeqyYdCNaey4kaSIaeqOSdigaaa@4546@ where α size 12{α} {} and β size 12{β} {}are constants. This case is not strictly linear phase as in Equation 5.15a , and is called generalized linear phase.To be specific , for β=0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIjabg2da9iaaicdaaaa@3948@ or π MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@37A4@ we have linear phase, for other values of β MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIbaa@3788@ such as π/2 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeGabiGaciaacaqaaeaadaqaaqaaaOqaamaalyaabaGaeqiWdahabaGaaGOmaaaaaaa@3874@ we have generalized linear phase(see ).

Group delay The derivative of the phase with respect to frequency also has a meanning of delay and is called group delay or envelope delay, denoted τg(ω) size 12{τ rSub { size 8{g} } \( ω \) } {}: τ g (ω)  =  - dΦ(ω) dω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes8a0naaBaaaleaacaWGNbaabeaakiaacIcacqaHjpWDcaGGPaGaaGjbVpaaxacabaGaeyypa0daleqabaaaaOGaaGjbVlaac2cadaWcaaqaaiaadsgacqqHMoGrcaGGOaGaeqyYdCNaaiykaaqaaiaadsgacqaHjpWDaaaaaa@4965@ For linear phase in Equation (5.15a), the group delay is τ g ( ω ) = α size 12{τ rSub { size 8{g} } \( ω \) =α} {} which is also constant, the same as for phase delay in Equation (5.16). The idea is when the input signal contains many sinusoidal components which are not harmonically related, the phase delay is used together with the group delay to fully account for the phase change of the input signal . For the generalized linear phase Equation (5.17) the phase delay is τ p (ω)=− αω+β ω =α− β ω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeGabiGaciaacaqaaeaadaqaaqaaaOqaaiabes8a0naaBaaaleaacaWGWbaabeaakiaacIcacqaHjpWDcaGGPaGaeyypa0JaeyOeI0YaaSaaaeaacqaHXoqycqaHjpWDcqGHRaWkcqaHYoGyaeaacqaHjpWDaaGaeyypa0JaeqySdeMaeyOeI0YaaSaaaeaacqaHYoGyaeaacqaHjpWDaaaaaa@4CC9@ dependent on frequency ω size 12{ω} {}, whereas the group delay is τ g (ω)=α MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeGabiGaciaacaqaaeaadaqaaqaaaOqaaiabes8a0naaBaaaleaacaWGNbaabeaakiaacIcacqaHjpWDcaGGPaGaeyypa0JaeqySdegaaa@3E96@ independent of frequency ω size 12{ω} {}. Consider for example a filter having transfer function H(z)=z−2 size 12{H \( z \) =z rSup { size 8{ - 2} } } {}. Remember that this just means the filter delays the input signal x(n) size 12{x \( n \) } {} two time indices (samples) to give the output signal x(n−2) size 12{x \( n - 2 \) } {}. The frequency response is obtained by the replacement z= e jω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadQhacqGH9aqpcaWGLbWaaWbaaSqabeaacaWGQbGaeqyYdChaaaaa@3BBA@ , thus H ( ω ) = e − j2ω size 12{H \( ω \) =e rSup { size 8{ - j2ω} } } {} The magnitude and phase spectra are , respectively, ∣ H ( ω ) ∣ = 1 size 12{ lline H \( ω \) rline =1} {} Φ ( ω ) = − 2ω size 12{Φ \( ω \) = - 2ω} {} The phase spectrum is shown in . The actual plot of the phase −2ω size 12{ - 2ω} {} in the period −π,π size 12{ left [ - π,π right ]} {} is the straight line AB , but, by convention, the phase spectrum variation is limited to the range −π,π size 12{ left [ - π,π right ]} {}. This matter has been discussed many times in chapter 3.

Phase spectrum of H(ω)= e −j2ω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiaadIeacaGGOaGaeqyYdCNaaiykaiabg2da9iaadwgadaahaaWcbeqaaiabgkHiTiaadQgacaaIYaGaeqyYdChaaaaa@4077@

Types of linear phase filters For causal FIR filters described by Equation (5.2), depending on whether the order M is even or odd, and whether the impulse response (filter coefficients) h(n) size 12{h \( n \) } {} is symmetric (also called even-symmetric or positive-symmetric) or antisymmetric (also called odd-symmetric or negative-symmetric), they are divided into four different types having different characteristics . Fig.5.4 depicts the four types. In the following we consider the causal FIR filter of Equation (5.2). FIR-1 The filter order M is even , and its impulse response is symmetric (a) , i.e. h( n )=h( M-n ) , 0≤n≤M    0 ,     otherwise MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiAamaabmaabaGaamOBaaGaayjkaiaawMcaaiabg2da9iaadIgadaqadaqaaiaad2eacaGGTaGaamOBaaGaayjkaiaawMcaaiaaywW7caGGSaGaaGzbVlaaicdacqGHKjYOcaWGUbGaeyizImQaamytaaqaaiaaywW7caaMf8UaaGzbVlaaicdacaaMf8UaaiilaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaam4BaiaadshacaWGObGaamyzaiaadkhacaWG3bGaamyAaiaadohacaWGLbaaaaa@60ED@ The above response is valid for 0≤n≤M MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaaicdacqGHKjYOcaWGUbGaeyizImQaamytaaaa@3BCC@ but due to the symmetry, the actual range is 0≤n≤M/2 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaaicdacqGHKjYOcaWGUbGaeyizImQaamytaiaac+cacaaIYaaaaa@3D3B@ . The 4 cases of impulse response in Fig.3.31 all belong to FIR-1. Let’s first consider a simple example with M=4 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eacqGH9aqpcaaI0aaaaa@387C@ . Noticing that h(0)=h(4) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgacaGGOaGaaGimaiaacMcacqGH9aqpcaWGObGaaiikaiaaisdacaGGPaaaaa@3CF0@ , h(1)=h(3) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgacaGGOaGaaGymaiaacMcacqGH9aqpcaWGObGaaiikaiaaiodacaGGPaaaaa@3CF0@ , and h(2) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgacaGGOaGaaGOmaiaacMcaaaa@38E8@ is by itself, we write the expression for frequency response as   H(ω)=h(0)+h(1) e −jω +h(2) e −j2ω +h(1) e −j3ω +h(0) e −j4ω       = e −j2ω [ h( 2 )+h( 0 )( e j2ω + e −j2ω )+h( 1 )( e jω + e −jω ) ]       = e −j2ω [ h( 2 )+2h( 0 )cos⁡2ω+2h( 1 )cos⁡ω ] or   H(ω)= e −j2ω [ h( 2 )+2 ∑ k=0 1 h(k)cos⁡( 2−k )ω ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@EB40@ For the filter of order M it can be shown that H(ω) =  e -jωM/2 [ h( M 2 )+2 ∑ k=0 M 2 -1 h(k)cos⁡ω(M/2-k) ] (FIR−1) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIeacaGGOaGaeqyYdCNaaiykaiaaysW7cqGH9aqpcaaMe8UaamyzamaaCaaaleqabaGaaiylaiaadQgacqaHjpWDcaWGnbGaai4laiaaikdaaaGcdaWadaqaaiaadIgacaGGOaWaaSaaaeaacaWGnbaabaGaaGOmaaaacaGGPaGaey4kaSIaaGOmamaaqahabaGaamiAaiaacIcacaWGRbGaaiykaiGacogacaGGVbGaai4CaiabeM8a3jaacIcacaWGnbGaai4laiaaikdacaGGTaGaam4AaiaacMcaaSqaaiaadUgacqGH9aqpcaaIWaaabaWaaSGaaeaacaWGnbaabaGaaGOmaaaacaGGTaGaaGymaaqdcqGHris5aaGccaGLBbGaayzxaaGaaGzbVlaacIcacaWGgbGaamysaiaadkfacqGHsislcaaIXaGaaiykaaaa@6853@

Phase spectrum of

which is of the form H(ω)= e −jωM/2 G(ω) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIeacaGGOaGaeqyYdCNaaiykaiabg2da9iaadwgadaahaaWcbeqaaiabgkHiTiaadQgacqaHjpWDcaWGnbGaai4laiaaikdaaaGccaWGhbGaaiikaiabeM8a3jaacMcaaaa@45DC@ where G(ω) size 12{G \( ω \) } {} is real but can be positive or negative. When G(ω)>0 size 12{G \( ω \) >0} {} the phase delay is and the filter is strictly linear phase. When G(ω)<0 size 12{G \( ω \) <0} {} the phase is Φ(ω)=− M 2 ω+π MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabfA6agjaacIcacqaHjpWDcaGGPaGaeyypa0JaeyOeI0YaaSaaaeaacaWGnbaabaGaaGOmaaaacqaHjpWDcqGHRaWkcqaHapaCaaa@4283@ In this case, β MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiabek7aIbaa@37A3@ in Equation (5.17) is π MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiabec8aWbaa@37BF@ . It seems that there will be phase distortion (i.e. signal waveshape is distorted). Fortunately the amplitude of G(ω) size 12{G \( ω \) } {} and hence of H(ω) size 12{H \( ω \) } {} can be negative only in the stopband where the magnitude is quite small (compared to that in the passband) hence the effect of distortion is usually acceptable. All the basic frequency selective filters (Fig.5.1) can be of FIR type 1 when M is even (see Fig.3.31) FIR-2 The filter order M is odd , and its impulse response is symmetric (b) as in Equation (5.20) but the actual range is 0≤n≤(M−1)/2 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaaicdacqGHKjYOcaWGUbGaeyizImQaaiikaiaad2eacqGHsislcaaIXaGaaiykaiaac+cacaaIYaaaaa@403C@ It can be shown that the frequency response is H( ω ) = 2 e -jωM/2 ∑ k=0 (M-1)/2 h( k )cos⁡ω( M/2 -k )  (FIR−2) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIeadaqadaqaaiabeM8a3bGaayjkaiaawMcaaiaaysW7cqGH9aqpcaaMe8UaaGOmaiaadwgadaahaaWcbeqaaiaac2cacaWGQbGaeqyYdC3aaSGbaeaacaWGnbaabaGaaGOmaaaaaaGcdaaeWbqaaiaadIgadaqadaqaaiaadUgaaiaawIcacaGLPaaaciGGJbGaai4BaiaacohacqaHjpWDdaqadaqaamaalyaabaGaamytaaqaaiaaikdaaaGaaiylaiaadUgaaiaawIcacaGLPaaaaSqaaiaadUgacqGH9aqpcaaIWaaabaGaaiikaiaad2eacaGGTaGaaGymaiaacMcacaGGVaGaaGOmaaqdcqGHris5aOGaaGzbVlaacIcacaWGgbGaamysaiaadkfacqGHsislcaaIYaGaaiykaaaa@62EC@ The phase linearity is the same as type 1 . Any of the basic frequency selective (Fig.5.1) can be of FIR type 2 when M is odd. FIR-3 The filter order M is even , and its impulse response is antisymmetric (c), i.e. h( n ) = -h( M-N ) , 0≤n≤M     0 ,      otherwise MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiAamaabmaabaGaamOBaaGaayjkaiaawMcaaiaaysW7cqGH9aqpcaaMe8UaaiylaiaadIgadaqadaqaaiaad2eacaGGTaGaamOtaaGaayjkaiaawMcaaiaaywW7caGGSaGaaGzbVlaaicdacqGHKjYOcaWGUbGaeyizImQaamytaaqaaiaaywW7caaMf8UacGzbVlaaysW7caaIWaGaaGzbVlaacYcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaaysW7caWGVbGaamiDaiaadIgacaWGLbGaamOCaiaadEhacaWGPbGaam4Caiaadwgaaaaa@67B3@ The above response is valid for 0≤n≤M MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaaicdacqGHKjYOcaWGUbGaeyizImQaamytaaaa@3BCC@ , but, due to the antisymmetry, the actual range is 0≤n≤M MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaaicdacqGHKjYOcaWGUbGaeyizImQaamytaaaa@3BCC@ as in FIR-1. The frequency response can be shown to be H( ω ) = 2 e j( -ωM/2 +π/2 ) ∑ k=0 M/2-1 h( k )sin⁡ω( M/2 -k )  (FIR−3) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6693@ Notice that the filter introduces an additional phase advance of π/2 size 12{π/2} {} (which is β MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeGabiGaciaacaqaaeaadaqaaqaaaOqaaiabek7aIbaa@3786@ in Equation (5.17)), constant with respect to frequency. The phase response is Φ(ω)=− M 2 ω+ π 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfA6agjaacIcacqaHjpWDcaGGPaGaeyypa0JaeyOeI0YaaSaaaeaacaWGnbaabaGaaGOmaaaacqaHjpWDcqGHRaWkdaWcaaqaaiabec8aWbqaaiaaikdaaaaaaa@434E@ i.e. generalized linear phase. The digital differentiator and the digital Hilbert transformer (Fig.5.2) can be of FIR type 3 when M is even (see Fig.5.8). FIR-4 The filter order M is odd , and its impulse response is antisymmetric (d) as in Equation (5.24) but the actual range is 0≤n≤(M−1)/2 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaaicdacqGHKjYOcaWGUbGaeyizImQaaiikaiaad2eacqGHsislcaaIXaGaaiykaiaac+cacaaIYaaaaa@403C@ as in the FIR-2. The frequency response can be shown to be H( ω ) = 2 e j( -ωM/2 +π/2 ) ∑ k=0 (M-1)/2 h( k )sin⁡ω( M/2 -k )  (FIR−4) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@67ED@ The phase linearity is the same as type 3 above . The digital differentiator can be of FIR type 4 when M is odd. For types 3 and 4, both antisymmetric, the frequency response is always zero at frequency ω=0 size 12{ω=0} {}, thus they cannot be used as lowpass filters. On the other hand, their addition of π/2 size 12{π/2} {} phase shift make them useful in the design of differentiators and Hilberl transformers (Fig.5.2). Type 1 and 2 are more versatile. To be more radical, we examine the transfer functions instead of the frequency responses. For linear phase filters, as discussed previously , their impulse responses must be either symmetric or antisymmetric,i.e. h(n)=±h(M−n) size 12{h \( n \) = +- h \( M - n \) } {} which is transformed to H ( z ) = ∑ n = 0 M h ( n ) z − n = ± ∑ n = 0 M h ( M − n ) z − n size 12{H \( z \) = Sum cSub { size 8{n=0} } cSup { size 8{M} } {h \( n \) z rSup { size 8{ - n} } = +- Sum cSub { size 8{n=0} } cSup { size 8{M} } {h \( M - n \) z rSup { size 8{ - n} } } } } {} A change of variable will lead to H( z )=± z -M H( z -1 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIeadaqadaqaaiaadQhaaiaawIcacaGLPaaacqGH9aqpcqGHXcqScaWG6bWaaWbaaSqabeaacaGGTaGaamytaaaakiaadIeadaqadaqaaiaadQhadaahaaWcbeqaaiaac2cacaaIXaaaaaGccaGLOaGaayzkaaaaaa@43DC@ From here we can make various observations about the zero locations, for example for antisymmetric filters, both even and odd M, there is a zero at z=1 size 12{z=1} {} ; for the case of M even there is an additional zero at z=−1 size 12{z= - 1} {} (making the filter suitable for bandpass filter). When the summation for H(ω) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIeacaGGOaGaeqyYdCNaaiykaaaa@39D6@ and H(z) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIeacaGGOaGaamOEaiaacMcaaaa@3908@ is from n=0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaad6gacqGH9aqpcaaIWaaaaa@3896@ to n−1 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaad6gacqGHsislcaaIXaaaaa@387E@ (as adopted by many other authors), instead of n−1 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaad6gacqGHsislcaaIXaaaaa@387E@ to M as we use here, then in all previous expressions involuing M we should replace N – 1 by M (or N by M – 1). From the known impulse response of an ideal lowpass filter having cutoff frequency ωc size 12{ω rSub { size 8{c} } } {}, deduce the impulse response for the causal linear phase filter having M coefficients.

Example (impulse response of ideal lowpass filter for ω c =π/2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiabeM8a3naaBaaaleaacaWGJbaabeaakiabg2da9maalyaabaGaeqiWdahabaGaaGOmaaaaaaa@3C82@ ) and M=4 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiaad2eacqGH9aqpcaaI0aaaaa@3898@

Solution The impulse response of an ideal lowpass filter has been found in Example 3.7.2 to be h LP (n)= sin⁡ ω c n πn = ω c π ⋅ sin⁡ ω c n ω c n  , n≠0        ω c π  ,        n=0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@7ED4@

Example continued (Matlab simulation for the case ω c =π/ 2 , M=4 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiabeM8a3naaBaaaleaacaWGJbaabeaakiabg2da9maalyaabaGaeqiWdahabaGaaGOmaiaaysW7caGGSaGaaGjbVlaad2eacqGH9aqpcaaI0aaaaaaa@42E2@ )

Now let’s truncate the impulse response at −N size 12{ - N} {}and N size 12{N} {}, whereby retaining 2N+1 size 12{2N+1} {} samples, then shift the truncated sequence to the right N size 12{N} {} samples so that the previous sample at n=−N size 12{n= - N} {} is now at the orgin n=0 size 12{n=0} {}, the previous sample at N size 12{N} {} is now at 2N size 12{2N} {} which is M size 12{M} {} in our notation, the previous sample at the origin is now at M/2 size 12{M/2} {}. shows the case M=4 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eacqGH9aqpcaaI0aaaaa@387B@ , the center of symmetry is at n=2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaad6gacqGH9aqpcaaIYaaaaa@389B@ . This is linear phase type 1 FIR filter. On replacing n by n−M/2 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaad6gacqGHsisldaWcgaqaaiaad2eaaeaacaaIYaaaaaaa@396B@ in above two-side impulse response we get the causal impulse response h LP (n)= sin⁡ ω c ( n−M/2 ) π( n−M/2 )  , 0≤n≤M , n≠M/2     ω c π  ,     n=M/2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiAamaaBaaaleaacaWGmbGaamiuaaqabaGccaGGOaGaamOBaiaacMcacqGH9aqpdaWcaaqaaiGacohacaGGPbGaaiOBaiabeM8a3naaBaaaleaacaWGJbaabeaakmaabmaabaGaamOBaiabgkHiTmaalyaabaGaamytaaqaaiaaikdaaaaacaGLOaGaayzkaaaabaGaeqiWda3aaeWaaeaacaWGUbGaeyOeI0YaaSGbaeaacaWGnbaabaGaaGOmaaaaaiaawIcacaGLPaaaaaGaaGjbVlaacYcacaaMf8UaaGimaiabgsMiJkaad6gacqGHKjYOcaWGnbGaaGjbVlaacYcacaaMf8UaamOBaiabgcMi5oaalyaabaGaamytaaqaaiaaikdaaaaabaGaaGzbVlaaywW7caaMf8+aaSaaaeaacqaHjpWDdaWgaaWcbaGaam4yaaqabaaakeaacqaHapaCaaGaaGjbVlaacYcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaad6gacqGH9aqpdaWcgaqaaiaad2eaaeaacaaIYaaaaaaaaa@7690@ This result can also be obtained by noticing that the frequency response of an ideal causal lowpass filter having impulse response symmetrical about n=M/2 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaad6gacqGH9aqpdaWcgaqaaiaad2eaaeaacaaIYaaaaaaa@3984@ is H LP (ω)= e −j M 2 ω  , −π≤− ω c ≤ω≤ ω c <π     0 ,     otherwise MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamisamaaBaaaleaacaWGmbGaamiuaaqabaGccaGGOaGaeqyYdCNaaiykaiabg2da9iaadwgadaahaaWcbeqaaiabgkHiTiaadQgadaWcaaqaaiaad2eaaeaacaaIYaaaaiabeM8a3baakiaaywW7caGGSaGaaGzbVlabgkHiTiabec8aWjabgsMiJkabgkHiTiabeM8a3naaBaaaleaacaWGJbaabeaakiabgsMiJkabeM8a3jabgsMiJkabeM8a3naaBaaaleaacaWGJbaabeaakiabgYda8iabec8aWbqaaiaaywW7caaMf8UaaGzbVlaaywW7caaIWaGaaGzbVlaacYcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaad+gacaWG0bGaamiAaiaadwgacaWGYbGaam4DaiaadMgacaWGZbGaamyzaaaaaa@738E@ The impulse response is the inverse DTFT of this frequency response : h LP = 1 2π ∫ −π π e −j M 2 ω e jωn dω= 1 2π ∫ − ω c ω c e −j(n− M 2 )ω dω MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6AE8@ Taking the integration we will get the result as in Equation (5.32). shows the Matlab simulation of the filter.■ Find the impulse response for −10≤n≤10 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabgkHiTiaaigdacaaIWaGaeyizImQaamOBaiabgsMiJkaaigdacaaIWaaaaa@3E17@ of a bandpass filter having lower cutoff frequency ω l = 2π /5 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3naaBaaaleaacaWGSbaabeaakiabg2da9maalyaabaGaaGOmaiabec8aWbqaaiaaiwdaaaaaaa@3D2B@ and upper cutoff frequency ω u = 3π /5 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3naaBaaaleaacaWG1baabeaakiabg2da9maalyaabaGaaG4maiabec8aWbqaaiaaiwdaaaaaaa@3D35@ . Deduce Find the frequency response from the truncated impulse response . Repeat for −50≤n≤50 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeGabiGaciaacaqaaeaadaqaaqaaaOqaaiabgkHiTiaaiwdacaaIWaGaeyizImQaamOBaiabgsMiJkaaiwdacaaIWaaaaa@3E21@ . solution The ideal bandpass filter is given in Fig.5.1c and general filter specifications are depicted in Fig.5.9. Its infinite duration impulse response is from the Equation (5.10b): h BP (n)  = sin⁡ ω u n πn − sin⁡ ω l n πn  , n≠0         ω u π − ω l π  ,     n=0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@7C4C@ a shows the ideal impulse response for −10≤n≤10 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabgkHiTiaaigdacaaIWaGaeyizImQaamOBaiabgsMiJkaaigdacaaIWaaaaa@3E17@ (that is M/ 2=10 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaamaalyaabaGaamytaaqaaiaaikdacqGH9aqpcaaIXaGaaGimaaaaaaa@3A02@ or M=20 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaad2eacqGH9aqpcaaIYaGaaGimaaaa@3931@ ). Notice that the filter is linear phase FIR.1. The frequency response of the designed filter is given by H(ω)= ∑ n=−10 10 h(n) e −jωn  , −π≤ω≤π     =h(0)+2 ∑ n=1 10 h(n) cos⁡nω MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@7462@ b plots |H(ω)| MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacYhacaWGibGaaiikaiabeM8a3jaacMcacaGG8baaaa@3BD6@ and c plots |H(ω) | dB MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacYhacaWGibGaaiikaiabeM8a3jaacMcacaGG8bWaaSbaaSqaaiaadsgacaWGcbaabeaaaaa@3DB2@ . The magnitude response is too far from ideal. We can start with a causal filter and use for the frequency response.

Example (linear phase bandpass filter with ω l = 2π /5  ,  ω u = 3π /5 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiabeM8a3naaBaaaleaacaWGSbaabeaakiabg2da9maalyaabaGaaGOmaiabec8aWbqaaiaaiwdaaaGaaGjbVlaacYcacaaMe8UaeqyYdC3aaSbaaSqaaiaadwhaaeqaaOGaeyypa0ZaaSGbaeaacaaIZaGaeqiWdahabaGaaGynaaaaaaa@4866@ ; and (b), (c) M=20 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiaad2eacqGH9aqpcaaIYaGaaGimaaaa@3950@ ; (d), (e) M=100 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiaad2eacqGH9aqpcaaIXaGaaGimaiaaicdaaaa@3A09@ )

The case −50≤n≤50 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeGabiGaciaacaqaaeaadaqaaqaaaOqaaiabgkHiTiaaiwdacaaIWaGaeyizImQaamOBaiabgsMiJkaaiwdacaaIWaaaaa@3E21@ is proceeded as above, the result is shown is d and e.The frequency response is much nearer to that of the ideal filter , but the ripples are still quite noticeable.■