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 ) 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) 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 ) ∠ Y ( ω ) =∠ H ( ω ) +∠ X ( ω ) size 12{∠Y \( ω \) "=∠"H \( ω \) "+∠"X \( ω \) } {}

By the above phase relation , the phase ∠H(ω)∠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 T 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsfadaWgaaWcbaGaaGimaaqabaaaaa@37A1@ sec, and angular frequency ω0ω0 size 12{ω rSub { size 8{0} } } {} rad/sec, ( ω 0 =2π/ T 0 ) ( 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 2π2π size 12{2π} {} rad, a phase delay of say Φ0Φ0 size 12{Φ rSub { size 8{0} } } {} will correspond to a time delay of τp=(Φ0/2π)T0τp=(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)Φ(ω) size 12{Φ \( ω \) } {} divided by the corresponding frequency ωω size 12{ω} {}:

Notice that even called phase delay but τp(ω)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:

This means

where αα size 12{α} {} is a constant and H 0 H 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIeadaWgaaWcbaGaaGimaaqabaaaaa@379A@ is a gain factor independent of frequency . If αα size 12{α} {} is possitive the filter delays the input signal, if negative it advances the input signal. The phase delay is now

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

where αα size 12{α} {} and ββ 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 β=0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIjabg2da9iaaicdaaaa@3948@ or π π MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@37A4@ we have linear phase, for other values of β β MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIbaa@3788@ such as π/2 π/2 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeGabiGaciaacaqaaeaadaqaaqaaaOqaamaalyaabaGaeqiWdahabaGaaGOmaaaaaaa@3874@ we have generalized linear phase(see (Reference)).

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)τg(ω) size 12{τ rSub { size 8{g} } \( ω \) } {}:

For linear phase in Equation (5.15a), the group delay is

τ g ( ω ) = α 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

dependent on frequency ωω size 12{ω} {}, whereas the group delay is

τ g (ω)=α τ g (ω)=α MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeGabiGaciaacaqaaeaadaqaaqaaaOqaaiabes8a0naaBaaaleaacaWGNbaabeaakiaacIcacqaHjpWDcaGGPaGaeyypa0JaeqySdegaaa@3E96@

independent of frequency ωω size 12{ω} {}.

Consider for example a filter having transfer function H(z)=z−2H(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)x(n) size 12{x \( n \) } {} two time indices (samples) to give the output signal x(n−2)x(n−2) size 12{x \( n - 2 \) } {}. The frequency response is obtained by the replacement z= e jω z= e jω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadQhacqGH9aqpcaWGLbWaaWbaaSqabeaacaWGQbGaeqyYdChaaaaa@3BBA@ , thus

H ( ω ) = e − j2ω H ( ω ) = e − j2ω size 12{H \( ω \) =e rSup { size 8{ - j2ω} } } {}

The magnitude and phase spectra are , respectively,

∣ H ( ω ) ∣ = 1 ∣ H ( ω ) ∣ = 1 size 12{ lline H \( ω \) rline =1} {}

Φ ( ω ) = − 2ω Φ ( ω ) = − 2ω size 12{Φ \( ω \) = - 2ω} {}

The phase spectrum is shown in Figure 1. The actual plot of the phase −2ω−2ω size 12{ - 2ω} {} in the period −π,π−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−π,π size 12{ left [ - π,π right ]} {}. This matter has been discussed many times in chapter 3.

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)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).

The filter order M is even , and its impulse response is symmetric (Figure 2a) , i.e.

The above response is valid for 0≤n≤M 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 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 M=4 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eacqGH9aqpcaaI0aaaaa@387C@ . Noticing that h(0)=h(4) h(0)=h(4) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgacaGGOaGaaGimaiaacMcacqGH9aqpcaWGObGaaiikaiaaisdacaGGPaaaaa@3CF0@ , h(1)=h(3) h(1)=h(3) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgacaGGOaGaaGymaiaacMcacqGH9aqpcaWGObGaaiikaiaaiodacaGGPaaaaa@3CF0@ , and h(2) 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

For the filter of order M it can be shown that

which is of the form

where G(ω)G(ω) size 12{G \( ω \) } {} is real but can be positive or negative. When G(ω)>0G(ω)>0 size 12{G \( ω \) >0} {} the phase delay is and the filter is strictly linear phase. When G(ω)<0G(ω)<0 size 12{G \( ω \) <0} {} the phase is

In this case, β β MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaaiaacaqaaeaadaqaaqaaaOqaaiabek7aIbaa@37A3@ in Equation (5.17) is π 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)G(ω) size 12{G \( ω \) } {} and hence of H(ω)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)

The filter order M is odd , and its impulse response is symmetric (Figure 2b) as in Equation (5.20) but the actual range is 0≤n≤(M−1)/2 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

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.

The filter order M is even , and its impulse response is antisymmetric (Figure 2c), i.e.

The above response is valid for 0≤n≤M 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 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

Notice that the filter introduces an additional phase advance of π/2π/2 size 12{π/2} {} (which is β 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

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).

The filter order M is odd , and its impulse response is antisymmetric (Figure 2d) as in Equation (5.24) but the actual range is 0≤n≤(M−1)/2 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

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