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FIR FILTERS

Module by: Nguyen Huu Phuong

First of all, the input-output signal difference equation of FIR filters is (Equation (2.18))
y(n)=∑k=−MMh(k)x(n−k)y(n)=∑k=−MMh(k)x(n−k) size 12{y \( n \) = Sum cSub { size 8{k= - M} } cSup { size 8{M} } {h \( k \) x \( n - k \) } } {}(1)
Where h(k) h(k) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgacaGGOaGaam4AaiaacMcaaaa@3918@ are coefficients , also the filter impulse response (Equation (2.19)), x(n−k) x(n−k) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiabgkHiTiaadUgacaGGPaaaaa@3B08@ is the input signal delayed by k time indices. For causal FIR filters the equation becomes
y(n)= ∑ k=0 M h(k)x(n−k)   =h(0)x(n)+h(1)x(n−1)+h(2)x(n−2)+...+h(M)x(n−M) y(n)= ∑ k=0 M h(k)x(n−k)   =h(0)x(n)+h(1)x(n−1)+h(2)x(n−2)+...+h(M)x(n−M) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@7072@ (2)
In this form the filter order is M, the filter length (total number of coefficients) is M+1 M+1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2eacqGHRaWkcaaIXaaaaa@3851@ . Some authors use N instead of M, and write the upper limit as N−1 N−1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6eacqGHsislcaaIXaaaaa@385D@ so that the filter length is N (this writing conforms with Matlab). We should stick to the same writing to avoid confusion.
The frequency response is the discrete – time Fourier transform (DTFT) of the impulse response :
H(ω)=∑n=0∞h(n)e−jωnH(ω)=∑n=0∞h(n)e−jωn size 12{H \( ω \) = Sum cSub { size 8{n=0} } cSup { size 8{ infinity } } {h \( n \) e rSup { size 8{ - jωn} } } } {}(3)
Knowing the frequency response H(ω)H(ω) size 12{H \( ω \) } {} we take the inverve DTFT to obtain the impulse response h(n)h(n) size 12{h \( n \) } {}. For system analysis, design and implementation the transfer function (or system function) is more important. It is (Equation (4.4))
H(z)= ∑ n=0 M h(n) z −n H(z)= ∑ n=0 M h(n) z −n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIeacaGGOaGaamOEaiaacMcacqGH9aqpdaaeWbqaaiaadIgacaGGOaGaamOBaiaacMcacaWG6bWaaWbaaSqabeaacqGHsislcaWGUbaaaaqaaiaad6gacqGH9aqpcaaIWaaabaGaamytaaqdcqGHris5aaaa@460E@ (4)
Recal that (section 4.25) the frequency response H(ω) H(ω) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIeacaGGOaGaeqyYdCNaaiykaaaa@39D5@ is obtained from the transfer function H(z) H(z) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIeacaGGOaGaamOEaiaacMcaaaa@3907@ on replacing z by e jω e jω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwgadaahaaWcbeqaaiaadQgacqaHjpWDaaaaaa@39B5@ .

Advantages and disadvantages of FIR filters

FIR filters have some advantages over IIR ones, the two most obvious are
- FIR filters can be designed to have linear phase (section 5.2), thus the waveshape of input signals is preserved. This is required in many application areas : data communication, image processing, biomedicine…
- FIR filters are always stable since they are nonrecursive systems. Their transfer functions are not rational prolynomials, and, thus, only have zeros. Only when certain coefficient (impulse response sample) is infinite then the filter is unstable, but inreality this does not happen . The guaranteed stability makes FIR filters very useful in adaptive filtering.
On the other hand , FIR filters have some disadvantages such as
- FIR filters have many times more coefficients to achieve the same frequency response (ripple, transition width), thus they require more processing time and larger storage.
- FIR filters have no analog counterparts, hence we cannot make use of analog design techniques.

Types of ideal frequency selective filters

Based on their frequency characteristics (or frequency responses) filters are classified as lowpass, highpass, bandpass, and bandstop (bandsuppress) (Fig.5.1). Ideal filters have constant passband amplitude, zero stopband amplitude and abrupt transition (transition time equals to zero) . Notice that the classification is not based on the phase characteristic. In details , filter can be categorized as smoothing, narrowband, notch, comb, allpass, mininum phase…
Figure 1: Basic types of ideal frequency selective filters
We should remember that the spectrum, frequency as well as phase, of discrete-time signals and systems is 2π2π size 12{2π} {}-periodic with the central period usually taken as the interval −π,π−π,π size 12{ left [ - π,π right ]} {} or 0,2π0,2π size 12{ left [0,2π right ]} {}.
In passing, it’s worth mentioning two filters which may be referred to sometimes, the digital differentiator and the Hilbert transformer defined , respectively , as
H(ω)=jω , −π≤ω≤π  (ideal differentiator) H(ω)=jω , −π≤ω≤π  (ideal differentiator) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIeacaGGOaGaeqyYdCNaaiykaiabg2da9iaadQgacqaHjpWDcaaMf8UaaiilaiaaywW7cqGHsislcqaHapaCcqGHKjYOcqaHjpWDcqGHKjYOcqaHapaCcaaMf8UaaGzbVlaacIcacaWGPbGaamizaiaadwgacaWGHbGaamiBaiaaysW7caWGKbGaamyAaiaadAgacaWGMbGaamyzaiaadkhacaWGLbGaamOBaiaadshacaWGPbGaamyyaiaadshacaWGVbGaamOCaiaacMcaaaa@62B7@ (5)
and
H(ω)=−j , 0≤ω≤π  (ideal Hilbert transformer)       j , −π≤ω≤0 H(ω)=−j , 0≤ω≤π  (ideal Hilbert transformer)       j , −π≤ω≤0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamisaiaacIcacqaHjpWDcaGGPaGaeyypa0JaeyOeI0IaamOAaiaaywW7caGGSaGaaGzbVlaaicdacqGHKjYOcqaHjpWDcqGHKjYOcqaHapaCcaaMf8UaaGzbVlaacIcacaWGPbGaamizaiaadwgacaWGHbGaamiBaiaaysW7caWGibGaamyAaiaadYgacaWGIbGaamyzaiaadkhacaWG0bGaaGjbVlaadshacaWGYbGaamyyaiaad6gacaWGZbGaamOzaiaad+gacaWGYbGaamyBaiaadwgacaWGYbGaaiykaaqaaiaaywW7caaMf8UaaGjbVlaaywW7caaMe8UaaGjbVlaadQgacaaMf8UaaiilaiaaywW7cqGHsislcqaHapaCcqGHKjYOcqaHjpWDcqGHKjYOcaaIWaaaaaa@7BDD@ (6)
For the ideal differentiator, the frequency response is
∣ H ( ω ) ∣ = ω ∣ H ( ω ) ∣ = ω size 12{ lline H \( ω \) rline =ω} {}
Φ ( ω ) = π Φ ( ω ) = π size 12{Φ \( ω \) =π} {}
Figure 2: Frequency response of digital differentiator and Hilbert transformer.
and for the Hilbert transformer,
∣ H ( ω ) ∣ = 1 ∣ H ( ω ) ∣ = 1 size 12{ lline H \( ω \) rline =1} {}
Φ(ω)= π 2  , −π≤ω≤0     − π 2  , 0≤ω≤π Φ(ω)= π 2  , −π≤ω≤0     − π 2  , 0≤ω≤π MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeuOPdyKaaiikaiabeM8a3jaacMcacqGH9aqpdaWcaaqaaiabec8aWbqaaiaaikdaaaGaaGjbVlaacYcacaaMf8UaeyOeI0IaeqiWdaNaeyizImQaeqyYdCNaeyizImQaaGimaaqaaiaaywW7caaMf8UaaGzbVlaaysW7cqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaGjbVlaacYcacaaMf8UaaGimaiabgsMiJkabeM8a3jabgsMiJkabec8aWbaaaa@5FA4@
The results are shown in Fig.5.2.

Relations between basic ideal filters

From Fig.5.1 let’s first denote by hLP(n)hLP(n) size 12{h rSub { size 8{ ital "LP"} } \( n \) } {}and HLP(ω)HLP(ω) size 12{H rSub { size 8{ ital "LP"} } \( ω \) } {} the time and frequency characterization , respectively, of ideal lowpass filters. Then for frequency response the relations are
  • Highpass:
H HP (ω)=1− H LP (ω) H HP (ω)=1− H LP (ω) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiFCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIeadaWgaaWcbaGaamisaiaadcfaaeqaaOGaaiikaiabeM8a3jaacMcacqGH9aqpcaaIXaGaeyOeI0IaamisamaaBaaaleaacaWGmbGaamiuaaqabaGccaGGOaGaeqyYdCNaaiykaaaa@441D@ (7)
  • Bandpass:
H BP (ω)= H LP (ω) | ω u − H LP (ω) | ω l H BP (ω)= H LP (ω) | ω u − H LP (ω) | ω l MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIeadaWgaaWcbaGaamOqaiaadcfaaeqaaOGaaiikaiabeM8a3jaacMcacqGH9aqpcaWGibWaaSbaaSqaaiaadYeacaWGqbaabeaakiaacIcacqaHjpWDcaGGPaGaaiiFamaaBaaaleaacqaHjpWDdaWgaaadbaGaamyDaaqabaaaleqaaOGaeyOeI0IaamisamaaBaaaleaacaWGmbGaamiuaaqabaGccaGGOaGaeqyYdCNaaiykaiaacYhadaWgaaWcbaGaeqyYdC3aaSbaaWqaaiaadYgaaeqaaaWcbeaaaaa@518F@ (8)
  • Bandstop:
h BP (n)=δ(n)− h BP (n) h BP (n)=δ(n)− h BP (n) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgadaWgaaWcbaGaamOqaiaadcfaaeqaaOGaaiikaiaad6gacaGGPaGaeyypa0JaeqiTdqMaaiikaiaad6gacaGGPaGaeyOeI0IaamiAamaaBaaaleaacaWGcbGaamiuaaqabaGccaGGOaGaamOBaiaacMcaaaa@45DC@ (9)
where ω u ω u MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBaaaleaacaWG1baabeaaaaa@38D5@ is the upper cutoff frequency , and ω l ω l MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBaaaleaacaWGSbaabeaaaaa@38CC@ the lower cutoff frequency. By taking the inverse DTFT we obtain the impulse response relations:
  • Highpass: hHP(n)=δ(n)−hLP(n)hHP(n)=δ(n)−hLP(n) size 12{h rSub { size 8{ ital "HP"} } \( n \) =δ \( n \) - h rSub { size 8{ ital "LP"} } \( n \) } {}(5.8a)
  • Bandpass: h BP (n)= h LP (n) | ω u − h LP (n) | ω l h BP (n)= h LP (n) | ω u − h LP (n) | ω l MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgadaWgaaWcbaGaamOqaiaadcfaaeqaaOGaaiikaiaad6gacaGGPaGaeyypa0JaamiAamaaBaaaleaacaWGmbGaamiuaaqabaGccaGGOaGaamOBaiaacMcacaGG8bWaaSbaaSqaaiabeM8a3naaBaaameaacaWG1baabeaaaSqabaGccqGHsislcaWGObWaaSbaaSqaaiaadYeacaWGqbaabeaakiaacIcacaWGUbGaaiykaiaacYhadaWgaaWcbaGaeqyYdC3aaSbaaWqaaiaadYgaaeqaaaWcbeaaaaa@4F61@
  • Bandstop: h BP (n)=δ(n)− h BP (n) h BP (n)=δ(n)− h BP (n) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgadaWgaaWcbaGaamOqaiaadcfaaeqaaOGaaiikaiaad6gacaGGPaGaeyypa0JaeqiTdqMaaiikaiaad6gacaGGPaGaeyOeI0IaamiAamaaBaaaleaacaWGcbGaamiuaaqabaGccaGGOaGaamOBaiaacMcaaaa@45DC@
For example the impulse response of the ideal lowpass filter having cutoff frequency ω c ω c MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBaaaleaacaWGJbaabeaaaaa@38C3@ was found in Example 3.7.3 as
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 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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@80F9@
where −∞<n<∞ −∞<n<∞ MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabg6HiLkabgYda8iaad6gacqGH8aapcqGHEisPaaa@3CB1@ . Then
h HP (n)=δ(n)− sin⁡ ω c n πn h HP (n)=δ(n)− sin⁡ ω c n πn MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgadaWgaaWcbaGaamisaiaadcfaaeqaaOGaaiikaiaad6gacaGGPaGaeyypa0JaeqiTdqMaaiikaiaad6gacaGGPaGaeyOeI0YaaSaaaeaaciGGZbGaaiyAaiaac6gacqaHjpWDdaWgaaWcbaGaam4yaaqabaGccaWGUbaabaGaeqiWdaNaamOBaaaaaaa@4A4D@
meaning
h HP (n)=− sin⁡ ω c n πn  ,  n≠0     1− ω c π   ,  n=0 h HP (n)=− sin⁡ ω c n πn  ,  n≠0     1− ω c π   ,  n=0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiAamaaBaaaleaacaWGibGaamiuaaqabaGccaGGOaGaamOBaiaacMcacqGH9aqpcqGHsisldaWcaaqaaiGacohacaGGPbGaaiOBaiabeM8a3naaBaaaleaacaWGJbaabeaakiaad6gaaeaacqaHapaCcaWGUbaaaiaaywW7caGGSaGaaGzbVlaaywW7caWGUbGaeyiyIKRaaGimaaqaaiaaywW7caaMf8UaaGzbVlaaywW7caaIXaGaeyOeI0YaaSaaaeaacqaHjpWDdaWgaaWcbaGaam4yaaqabaaakeaacqaHapaCaaGaaGzbVlaaywW7caGGSaGaaGzbVlaaywW7caWGUbGaeyypa0JaaGimaaaaaa@6564@
h BP (n)= sin⁡ ω u n πn − sin⁡ ω l n πn  , n≠0       ω u π − ω l π  ,     n=0 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@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiAamaaBaaaleaacaWGcbGaamiuaaqabaGccaGGOaGaamOBaiaacMcacqGH9aqpdaWcaaqaaiGacohacaGGPbGaaiOBaiabeM8a3naaBaaaleaacaWG1baabeaakiaad6gaaeaacqaHapaCcaWGUbaaaiabgkHiTmaalaaabaGaci4CaiaacMgacaGGUbGaeqyYdC3aaSbaaSqaaiaadYgaaeqaaOGaamOBaaqaaiabec8aWjaad6gaaaGaaGzbVlaacYcacaaMf8UaamOBaiabgcMi5kaaicdaaeaacaaMf8UaaGzbVlaaywW7caaMe8UaaGjbVpaalaaabaGaeqyYdC3aaSbaaSqaaiaadwhaaeqaaaGcbaGaeqiWdahaaiabgkHiTmaalaaabaGaeqyYdC3aaSbaaSqaaiaadYgaaeqaaaGcbaGaeqiWdahaaiaaywW7caGGSaGaaGzbVlaaywW7caaMf8UaaGzbVlaaysW7caWGUbGaeyypa0JaaGimaaaaaa@7620@
h BS (n)=δ(n)−[ sin ω u n πn − sin⁡ ω l n πn ] , n≠0 h BS (n)=δ(n)−[ sin ω u n πn − sin⁡ ω l n πn ] , n≠0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIgadaWgaaWcbaGaamOqaiaadofaaeqaaOGaaiikaiaad6gacaGGPaGaeyypa0JaeqiTdqMaaiikaiaad6gacaGGPaGaeyOeI0YaamWaaeaadaWcaaqaaiaabohacaqGPbGaaeOBaiabeM8a3naaBaaaleaacaWG1baabeaakiaad6gaaeaacqaHapaCcaWGUbaaaiabgkHiTmaalaaabaGaci4CaiaacMgacaGGUbGaeqyYdC3aaSbaaSqaaiaadYgaaeqaaOGaamOBaaqaaiabec8aWjaad6gaaaaacaGLBbGaayzxaaGaaGzbVlaacYcacaaMf8UaamOBaiabgcMi5kaaicdaaaa@5DF5@
meaning
h BS (n)= sin⁡ ω l n πn − sin⁡ ω u n πn  ,  n≠0     1−( ω u π − ω l π )  ,  n=0 h BS (n)= sin⁡ ω l n πn − sin⁡ ω u n πn  ,  n≠0     1−( ω u π − ω l π )  ,  n=0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@760B@
Notice that for the same cutoff frequency ωcωc size 12{ω rSub { size 8{c} } } {} the lowpass filter and the highpass filter are complementary in the sense that the sum of their impulse responses is the unit sample δ(n)δ(n) size 12{δ \( n \) } {}. And for the same limiting frequencies ω l ω l MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeM8a3naaBaaaleaacaWGSbaabeaaaaa@38CC@ and ωuωu size 12{ω rSub { size 8{u} } } {}, the bandpass and bandstop filters are also complementary.

Transforming a lowpass filter to other filters

In previous section we mentioned the relations between ideal filters . In this section we discuss the transformation between general filters. Suppose the design of a lowpass filter has accomplished, i.e. its impulse response (filter coefficients) is known. By the use of the frequency shift property of the DTFT(section 3.5) we can transform the lowpass filter to other filters having the same main characteristics.
The frequency response of the highpass filter is obtained from the lowpass one by shifting the latter by ππ size 12{π} {} radians, i.e. by replacing ωω size 12{ω} {} by ω−πω−π size 12{ω - π} {}:
HHP(ω)=HLP(ω−π)HHP(ω)=HLP(ω−π) size 12{H rSub { size 8{ ital "HP"} } \( ω \) =H rSub { size 8{ ital "LP"} } \( ω - π \) } {}
Delaying the frequency of the frequency response by ππ size 12{π} {} radians corresponds to multiplying the impulse response by ejπnejπn size 12{e rSup { size 8{jπn} } } {}, thus the impulse response of the highpass counterpart is
h HP (n)= e jπn h LP (n) h HP (n)= e jπn h LP (n) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIgadaWgaaWcbaGaamisaiaadcfaaeqaaOGaaiikaiaad6gacaGGPaGaeyypa0JaamyzamaaCaaaleqabaGaamOAaiabec8aWjaad6gaaaGccaWGObWaaSbaaSqaaiaadYeacaWGqbaabeaakiaacIcacaWGUbGaaiykaaaa@45CE@
or
h HP (n)= (-1) n h LP (n)=(cos⁡πn) h LP (n) h HP (n)= (-1) n h LP (n)=(cos⁡πn) h LP (n) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaabIgadaWgaaWcbaGaaeisaiaabcfaaeqaaOGaciikaiaab6gaciGGPaGaaeypaiaacIcacaqGTaGaaeymaiaacMcadaahaaWcbeqaaiaab6gaaaGccaqGObWaaSbaaSqaaiaabYeacaqGqbaabeaakiGacIcacaqGUbGaciykaiabg2da9iaacIcaciGGJbGaai4BaiaacohacqaHapaCcaqGUbGaaiykaiaabIgadaWgaaWcbaGaaeitaiaabcfaaeqaaOGaciikaiaab6gaciGGPaaaaa@519B@
This result means we keep the sign of the even time indices (n=0,2,4...)(n=0,2,4...) size 12{ \( n=0,2,4 "." "." "." \) } {}of hLP(n)hLP(n) size 12{h rSub { size 8{ ital "LP"} } \( n \) } {}, and reverse the sign of the odd indices (n=1,3,5...)(n=1,3,5...) size 12{ \( n=1,3,5 "." "." "." \) } {}of hLP(n).hLP(n). size 12{h rSub { size 8{ ital "LP"} } \( n \) "." } {}The reverse transformation is also true:
h LP (n)= (-1) n h HP (n)=(cos⁡πn) h HP (n) h LP (n)= (-1) n h HP (n)=(cos⁡πn) h HP (n) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaabIgadaWgaaWcbaGaaeitaiaabcfaaeqaaOGaciikaiaab6gaciGGPaGaaeypaiGacIcacaqGTaGaaeymaiGacMcadaahaaWcbeqaaiaab6gaaaGccaqGObWaaSbaaSqaaiaabIeacaqGqbaabeaakiGacIcacaqGUbGaciykaiabg2da9iaacIcaciGGJbGaai4BaiaacohacaqGapGaaeOBaGqabiaa=LcacaqGObWaaSbaaSqaaiaabIeacaqGqbaabeaakiGacIcacaqGUbGaciykaaaa@512A@
In above , (−1)n(−1)n size 12{ \( - 1 \) rSup { size 8{n} } } {} is just cosnπcosnπ size 12{"cos"nπ} {}, we can write the impulse response in either of the two forms.
If the difference equation representing a lowpass filter , (Equation (2.21)), is
y(n)=∑k=1Naky(n−k)+∑k=−MMbkx(n−k)y(n)=∑k=1Naky(n−k)+∑k=−MMbkx(n−k) size 12{y \( n \) = Sum cSub { size 8{k=1} } cSup { size 8{N} } {a rSub { size 8{k} } y \( n - k \) + Sum cSub { size 8{k= - M} } cSup { size 8{M} } {b rSub { size 8{k} } x \( n - k \) } } } {}
then the equation of the highpass counterpart is
y(n)=∑k=1N(−1)kaky(n−k)+∑k=−MM(−1)kbkx(n−k)y(n)=∑k=1N(−1)kaky(n−k)+∑k=−MM(−1)kbkx(n−k) size 12{y \( n \) = Sum cSub { size 8{k=1} } cSup { size 8{N} } { \( - 1 \) rSup { size 8{k} } a rSub { size 8{k} } y \( n - k \) + Sum cSub { size 8{k= - M} } cSup { size 8{M} } { \( - 1 \) rSup { size 8{k} } b rSub { size 8{k} } x \( n - k \) } } } {}
Now let’s see how to transform a designed lowpass filter to a bandpass one. We know in DTFT that multiplying in time domain corresponds to convolution in frequency domain (section 3.5) and the DTFT of sinusoid cosnω0cosnω0 size 12{"cos"nω rSub { size 8{0} } } {} consists of two impulses at frequency ±ω0±ω0 size 12{ +- ω rSub { size 8{0} } } {} (section3.7.2 ). Thus when we multiply the impulse response of a lowpass filter by cosω0ncosω0n size 12{"cos"ω rSub { size 8{0} } n} {} we will obtain a corresponding bandpass filter having mid-frequency and bandwidth double that of the lowpass.

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