CN101442296A - Digital decimation filter - Google Patents

Abstract

The invention provides a digital decimation filter. The digital decimation filter comprises a CIC filter with a cascaded cosine prefilter, a compensating filter and an FIR filter which are connected sequentially. The digital decimation filter has the advantages that the digital decimation filter brings small passband landing, outputs small power of quantization noise; and the mode does not bring the increase of power consumption or brings very small increase of power consumption.

Description

A kind of decimation filter of digital

Technical field

The present invention relates to filter, relate in particular to a kind of decimation filter of digital.

Background technology

In the analog to digital converter (ADC) that decimation filter of digital (Digital Decimation Filter) is applied to modulating based on ∑-Δ.Analog to digital converter based on ∑-Δ modulation uses over-sampling, and noise shaped, decimation filter of digital obtains higher signal to noise ratio.Use over-sampling to be because along with the increase of multiple R, the overlapping component of quantization noise spectrum and signal spectra is fewer and feweri.The noise shaped noise energy that makes focuses on HFS.The digital lowpass decimation filter frequency band π/R＜| ω | remove quantization noise in the≤π, and signal component does not change, and has therefore improved signal to noise ratio.Decimation filter of digital not only can be used among the ADC that modulates based on ∑-Δ, can also be used in digital down converter (DDC) and the digital up converter (DUC).

The purpose of decimation filter of digital is to be f in order to make over-sampling speed _sDigital signal return to Nyquist speed f by filtering and down-sampling _NSignal, and keep simultaneously as far as possible only | ω | the signal in≤π/R, and leach out-of-band noise.It in fact be with regard to the cut-off frequency that serves as after the noise moulding | ω | the role of the low pass filter of≤π/R.R is called sample rate again and changes the factor, R=f _s/ f _NBy the Nyquist sampling thheorem, our interested analog signal peak frequency f then _c≤ f _N/ 2=f _s/ 2R is as normalization f _s/ 2 is π, then ω _c≤ π/R, wherein ω _c=2 π f _c

Cascaded integrator-comb (Cascaded Integrator-Comb is abbreviated as CIC) filter has characteristic of simple structure (when using the structure of integration-extraction-differential, not having multiplying), therefore generally uses the cic filter of low order at front end.Behind the process cic filter, speed decline R1 doubly.Than low rate the time, can use the interior fluctuation of passband little, the FIR filter that stopband attenuation is very fast.The tap number of such FIR filter is more, but owing to be operated under the low rate, therefore the increase of the complexity of bringing is little.

The cic filter on N rank is H (z)=((1-z ^-R)/(1-z ^-1)) ^N, wherein N is an exponent number.Using the reason of cic filter at the front end of decimation filter of digital is to realize simply, but does not have multiplication when using the IIR structure, and as N greatly the time, its side lobe attenuation is bigger.Generally speaking, the extraction factor R is 2 L power, i.e. R=2 ^L, then the transfer function of cic filter can be written as $H\left(z\right)={(1+z^{-1})}^N{(1+z^{-2})}^N\·\·\·{(1+z^{{-2}^{L-1}})}^N.$ Therefore cic filter can be implemented as the filtering-2 times extraction form of cascade.

General tolerance decimation filter performance has two indexs: (1) passband landing: the maximum attenuation of the interior filter of useful signal bandwidth (band of interest).(2) aliasing error: fold into the attenuation in the useful signal bandwidth.Aliasing error is of comparison key.Analog signal can be with sampling frequency f after being sampled _sFor periodically repeating frequency spectrum in the center, when sampling rate drops to f _s/ R then can be with f _s/ R is that frequency spectrum is periodically repeated at the center.Therefore after extracting R times, at [kf _s/ R-f _c, kf _s/ R+f _c] frequency band can be aliased in the interested frequency band, wherein As normalization f _s/ 2 is π, then [2k/R-ω _c, 2k/R+ ω _c] frequency band can be aliased in the interested frequency band, these frequency bands are called as the aliasing band.

The passband of cic filter descends bigger, and the increase that passband descends has caused reducing of bandwidth.Generally after extraction, increase the decline that a compensating filter compensates passband.The stopband attenuation of the cic filter of high-order is bigger, but the landing of passband is also bigger, causes that bandwidth reduces, and can make passband more smooth after compensating, but can amplify stopband simultaneously.Exponent number is high more, and it is many more that passband need compensate, and cause that the amplification of stopband is big more, and the passband fluctuation of the compensating filter of the cic filter of high-order is very big, causes that the fluctuation of passband of the cic filter after the compensation is also very big.The tap number that increases compensating filter can be improved the fluctuation of passband, but cost is the increase of complexity.

See also Fig. 1, existing decimation filter of digital comprises the cosine prefilter of cic filter and cascade.After existing digital decimation filtering is placed on cic filter with the cosine prefilter of cascade, but the shortcoming of existing digital decimation filtering is: it is not carried out the compensation of passband landing, so the passband landing is bigger.

Summary of the invention

The object of the present invention is to provide a kind of decimation filter of digital, be intended to solve the compensation of it not being carried out the passband landing of existing decimation filter of digital, the problem that the passband landing is bigger.

Technical scheme of the present invention is to realize like this, decimation filter of digital of the present invention comprises cic filter, compensating filter and the FIR filter of the cosine prefilter of being with cascade, and the cic filter of the cosine prefilter of described band cascade, compensating filter and FIR filter link to each other successively.

The technical scheme that the present invention takes also comprises: described compensating filter generates compensating filter by " frequency sampling method ".

The technical scheme that the present invention takes also comprises: the inverse of the frequency response of described compensating filter for being rung by the width of cloth of the filter that compensated.

The technical scheme that the present invention takes also comprises: described compensating filter is the compensating filter of 30 rank, 16 gains.

The technical scheme that the present invention takes also comprises: the cic filter of the cosine prefilter of the band cascade of described decimation filter of digital comprises the cosine prefilter of cic filter and band cascade.

The technical scheme that the present invention takes also comprises: described cascade cosine prefilter z variation is: $H_{\mathrm{CCOS}}\left(z\right)=\underset{i=1}{\overset{K}{\mathrm{\Π}}}{H}_{\mathrm{COS},i}\left(z^{N_i}\right),$

Wherein $H_{\mathrm{COS},i}\left(z\right)=1/4(z^{N_i}+z^{-N_i})+1/8(z^{2N_i}+z^{-2N_i})+1/4.$

The technical scheme that the present invention takes also comprises: described cascade cosine prefilter z variation is:

$H\left(z\right)=H_{\mathrm{CIC}}^4\left(z\right)H_{\mathrm{CIC}}^4\left(z^2\right)H_{\mathrm{CIC}}^4\left(z^4\right)H_{\mathrm{CIC}}^4\left(z^8\right)H_{\mathrm{CIC}}^4\left(z^{16}\right)H_{\mathrm{CCOS}}\left(z\right)$

H wherein _CIC(z)=1/2 (1+z ^-1),

$H_{\mathrm{CCOS}}\left(z\right)=H_{\mathrm{COS}}^{n_1}\left(z^8\right)H_{\mathrm{COS}}^{n_2}\left(z^4\right)H_{\mathrm{COS}}^{n_3}\left(z^2\right)H_{\mathrm{COS}}^{n_4}\left(z\right),$ n ₁＝n ₂＝n ₃＝n ₄＝4，

H _COS(z)＝1/4(z+z ^-1)+1/8(z ²+z ^-2)+1/4。

The technical scheme that the present invention takes also comprises: described cascade cosine prefilter z variation is:

$H\left(z\right)=H_{\mathrm{CIC}}^4\left(z\right)H_{\mathrm{CIC}}^4\left(z^2\right)H_{\mathrm{CIC}}^4\left(z^4\right)H_{\mathrm{CIC}}^4\left(z^8\right)H_{\mathrm{CCOS}}\left(z\right)$

H wherein _CIC(z)=1/2 (1+z ^-1), $H_{\mathrm{CCOS}}\left(z\right)=H_{\mathrm{COS}}^{n_1}\left(z^4\right)H_{\mathrm{COS}}^{n_2}\left(z^2\right)H_{\mathrm{COS}}^{n_3}\left(z\right),$ n ₁＝n ₂＝n ₃＝4，

H _COS(z)＝1/4(z+z ^-1)+1/8(z ²+z ^-2)+1/4。

Beneficial effect of the present invention is: decimation filter of digital introducing CIC of the present invention adds cascade cosine prefilter and adds the mode that compensation adds the FIR filter again, cascade cosine prefilter can strengthen stopband attenuation and not cause the big fluctuation of passband, compensation makes passband become more smooth, and the FIR filter reduces transition band length, the output quantization noise power of this mode is less, and the power consumption increase that this mode is brought is also little.

Feature of the present invention and advantage will be elaborated in conjunction with the accompanying drawings by embodiment.

Description of drawings

Fig. 1 is the structural representation of existing decimation filter of digital;

Fig. 2 is the structural representation of decimation filter of digital of the present invention;

Fig. 3 is the structural representation of cic filter of cosine prefilter of the band cascade of decimation filter of digital of the present invention;

Fig. 4 is the structural representation of the cosine prefilter of decimation filter of digital of the present invention;

Fig. 5 is the logarithm amplitude spectrum of CIC+FIR 4:1 mode and noise;

Fig. 6 is the logarithm amplitude spectrum of CIC+FIR 2:1 mode and noise;

Fig. 7 is the logarithm amplitude spectrum of CIC+CCOS+FIR 2:1 mode and noise;

Fig. 8 is the logarithm amplitude spectrum of CIC+CCOS2+FIR 4:1 mode and noise;

Fig. 9 is the passband part amplitude spectrum of the logarithm amplitude response of 4 rank CIC+CCOS1+FIR 2:1;

Figure 10 is the passband part amplitude spectrum of the logarithm amplitude response of 4 rank CIC+CCOS2+FIR 4:1;

Figure 11 is the passband part amplitude spectrum of the logarithm amplitude response of 4 rank CIC+FIR 4:1.

Embodiment

In order to make purpose of the present invention, technical scheme and advantage clearer,, the present invention is further elaborated below in conjunction with drawings and Examples.

See also Fig. 2, be the structural representation of decimation filter of digital of the present invention.Decimation filter of digital of the present invention comprises: cic filter, compensating filter and the FIR filter of the cosine prefilter of band cascade.Cic filter, compensating filter and the FIR filter of the cosine prefilter of band cascade link to each other successively.The present invention adopts " frequency sampling method " to generate compensating filter.The inverse of the frequency response of compensating filter for being rung, i.e. G (f)=1/|H (f) by the width of cloth of the filter that compensated |, generate compensating filter by " frequency sampling method " at last.

See also Fig. 3 and Fig. 4, the cic filter of the cosine prefilter of the band cascade of decimation filter of digital of the present invention comprises the cosine prefilter of cic filter and band cascade.

Cascade cosine prefilter (cascaded cosine prefilter), its z variation is:

$H_{\mathrm{CCOS}}\left(z\right)=\underset{i=1}{\overset{K}{\mathrm{\Π}}}{H}_{\mathrm{COS},i}\left(z^{N_i}\right)---\left(1\right)$

Wherein $H_{\mathrm{COS},i}\left(z\right)=1/4(z^{N_i}+z^{-N_i})+1/8(z^{2N_i}+z^{-2N_i})+1/4,$ Corresponding amplitude response is:

|H _cos，i(e ^jω)|＝1/2|cos(N _iω)+cos ²(N _iω)| (2)

n _iThe cosine prefilter on rank is designated as $H_{\mathrm{COS},i}^{n_i}\left(z\right)=\underset{i=1}{\overset{n_i}{\mathrm{\Π}}}{H}_{\mathrm{COS},i}\left(z\right),$ Then the cascade cosine prefilter of high-order is:

$H_{\mathrm{CCOS}}\left(z\right)=\underset{i=1}{\overset{K}{\mathrm{\Π}}}{H}_{\mathrm{COS},i}^{n_i}\left(z^{N_i}\right)---\left(3\right)$

N _iRepresented the stage that the cosine prefilter is placed, form is:

N _i＝R/2 ⁱ⁺¹ (4)

R=R wherein ₁R ₂... R _M, expression has M stage, and the extraction factor in each stage is R _iSelection to above-mentioned parameter among the present invention has two kinds of schemes:

(1) gets M=5, R ₁=R ₂=...=R ₅=2, R=32 then, and get N ₁=8, N ₂=4, N ₃=2, N ₄=1.

If H _CIC(z)=1/2 (1+z ^-1), then cascade the frequency response of cic filter of cosine prefilter be:

$H\left(z\right)=H_{\mathrm{CIC}}^4\left(z\right)H_{\mathrm{CIC}}^4\left(z^2\right)H_{\mathrm{CIC}}^4\left(z^4\right)H_{\mathrm{CIC}}^4\left(z^8\right)H_{\mathrm{CIC}}^4\left(z^{16}\right)H_{\mathrm{CCOS}}\left(z\right)---\left(5\right)$

Wherein $H_{\mathrm{CCOS}}\left(z\right)=H_{\mathrm{COS}}^{n_1}\left(z^8\right)H_{\mathrm{COS}}^{n_2}\left(z^4\right)H_{\mathrm{COS}}^{n_3}\left(z^2\right)H_{\mathrm{COS}}^{n_4}\left(z\right),$ Here get n ₁=n ₂=n ₃=n ₄=4.

We are designated as CIC+CCOS1 the cic filter of such cascade cosine prefilter.

(2) get M=4, R ₁=R ₂=R ₃=R ₄=2, R=16 then, and get N ₁=4, N ₂=2, N ₃=1.

If H _CIC(z)=1/2 (1+z ^-1), then cascade the frequency response of cic filter of cosine prefilter be:

$H\left(z\right)=H_{\mathrm{CIC}}^4\left(z\right)H_{\mathrm{CIC}}^4\left(z^2\right)H_{\mathrm{CIC}}^4\left(z^4\right)H_{\mathrm{CIC}}^4\left(z^8\right)H_{\mathrm{CCOS}}\left(z\right)---\left(6\right)$

Wherein $H_{\mathrm{CCOS}}\left(z\right)=H_{\mathrm{COS}}^{n_1}\left(z^4\right)H_{\mathrm{COS}}^{n_2}\left(z^2\right)H_{\mathrm{COS}}^{n_3}\left(z\right),$ Here get n ₁=n ₂=n ₃=4.

We are designated as CIC+CCOS2 the cic filter of such cascade cosine prefilter.

If H is (f _d) be the frequency response of whole decimation filter, signal and the total power spectral density of noise after then extracting are:

S _z(f _d)＝|H(f _d)| ²S _x(f _d)+|H(f _d)| ²S _N(f _d) (7)

S wherein _x(f _d) be the power spectral density of input signal, S _N(f _d) be the power spectral density of quantizing noise.Here we only discuss the power spectral density of quantizing noise.For the sigma-delta modulator on p rank, the power spectral density of quantizing noise is:

S _N(f _d)＝E(f _d)·[2sin(πf _d)] ^p (8)

Wherein $E\left(f_d\right)=({\mathrm{\Δ}}^2/12)\sqrt{2/f_s}$ For supposing that quantizing noise is the sampling noiset spectrum density of white noise, f _dNormalized frequency, corresponding angles frequency are ω=2 π f _d, f _d∈ [0,0.5], Δ=(2V _r/ 2 ^N-1) is the quantization step of N bit resolution quantizer, 2V _rDynamic range for the quantizer input.The transfer function of supposing quantizing noise is (1-z ^-1) ³, so the power spectral density of quantizing noise is:

$S_N\left(f_d\right)={\left|{(1-e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="A200710124780E00151.png,A200710124780E00152.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}{f}_d})}^3\right|}^2---\left(9\right)$

This is equivalent to E (f _d)=1, the situation during p=6.Below what carry out performance relatively is exactly in this case comparison.

The noise power spectral density of note output is below:

S _No(f _d)＝|H(f _d)| ²S _N(f _d) (10)

The noise power of output is (only calculating the power in the aliasing band):

$P_{\mathrm{xo}}=\underset{k}{\mathrm{\&Sigma;}}{\&Integral;}_{k/R-f_c}^{k/R+f_c}{S}_{\mathrm{No}}\left(f_d\right){\mathrm{df}}_d---\left(11\right)$

Wherein

For R is even number.

Suppose total extraction factor R=64, normalization f _s/ 2 is π, and then band of interest is at 0～π/R _ΔInterior (R _Δ≤ R), then the aliasing band is [2 π k/R-π/R _Δ, 2 π k/R+ π/R _Δ], wherein

R in general the application _Δ=R, so the aliasing band is in the whole interval of [0, π].Therefore in 0 to the π scope of decimation filter all the noise power to output exert an influence.

See also Fig. 5, Fig. 6, Fig. 7 and Fig. 8, compare 4 rank CIC+FIR 4:1 modes, 4 rank CIC+FIR 2:1 modes, 4 rank CIC+CCOS1+FIR 2:1 modes, the logarithm amplitude spectrum of the filter of 4 rank CIC+CCOS2+FIR4:1 modes respectively.In Fig. 5, Fig. 6, Fig. 7 and Fig. 8, empty vertical line is the starting point of aliasing band, and real vertical line is the terminal point of aliasing band.

See also Fig. 9, though the passband of 4 rank CIC+CCOS1+FIR 2:1 modes compensates a little excessive (the compensating filter exponent number increases and can improve), but compare fluctuation with 4 rank CIC+FIR 4:1 modes little, and the passband fluctuation of 8 rank or 10 rank CIC+FIR 4:1 modes is very big.See also Figure 10, the passband of 4 rank CIC+CCOS2+FIR 4:1 modes does not have fluctuation substantially, because the cic filter that uses will lack.Here used compensating filter all is 30 rank, 16 gains.

Following table is three kinds of noise powers that mode is estimated.

Table 1: the noise power of three kinds of modes relatively

	Word length of FIR	Pn(dB)
			CIC+FIR 4:1	24	-100
CIC+FIR 2:1	22	-94
			CIC+CCOS1+FIR 2:1	22	-153
CIC+CCOS2+FIR 4:1	24	-153

As seen the noise power of CIC+CCOS1+FIR 2:1 and CIC+CCOS2+FIR 4:1 mode is less.

Power consumption is estimated as following formula:

$P=\underset{i=1}{\overset{l}{\mathrm{\&Sigma;}}}\frac{{\mathrm{NP}}_i*W_i}{{\mathrm{\&Pi;}}_{j=1}^i{M}_j}---\left(12\right)$

NP wherein _iBe quantity in the partial product of i in the stage, W _iBe the input word length in i stage, M _jBe the extraction factor in i stage, l is the extraction stage sum.Here the partial product of saying is meant and the product of 2 power time, when multiplying each other with 2 power, just can be converted into integer and move to left when hardware is realized.For example when N=5, the FIR filter transfer function is (1+z ^-1) ⁵=1+5z ^-1+ 10z ^-2+ 10z ^-3+ 5z ^-4+ z ^-5, the impulse response of corresponding filter be [1 5 10 10 5 1], 2 corresponding systems are expressed as [0,001 0,101 1,010 1,010 0,101 0001], therefore have 10 times with the inferior product of 2 power, promptly have 10 times integer to move to left, corresponding NP _i=10.Because move to left each time with regard to a corresponding sub-addition, thus the actual number that moves to left with addition of having represented of the number of partial product, the complexity when it has well represented the hardware realization.Divided by

Be because the multiple that on behalf of speed, i stage total extraction number descend, and the default rate power consumption of one times of computing of being carried out one times of this fact that just descends that descends.

Compare CIC+FIR 4:1, CIC+FIR 2:1, the power consumption of CIC+CCOS1+FIR 2:1 and four kinds of modes of CIC+CCOS2+FIR 4:1 below.

The partial product quantity of the cic filter on 4 rank is 6, and word length is 4, and the partial product quantity of the compensating filter of the cic filter on 44 rank of compensation is 179, and word length is 16, and the partial product quantity of FIR 4:1 filter is 2945, and word length is 24.The partial product quantity of the compensating filter of the cic filter on 54 rank of compensation is 172, and word length is 16, and the partial product quantity of FIR 2:1 filter is 1383, and word length is 22.The partial product quantity of the cosine prefilter on 4 rank is 48, and word length is 3, and the partial product quantity that the cic filter on 54 rank of compensation adds the compensating filter of 44 rank cosine prefilters is 159, and word length is 16.The partial product quantity that the cic filter on 44 rank of compensation adds the compensating filter of 34 rank cosine prefilters is 110, and word length is 16.The input word length in i stage is W _i=W _I-1+ Wf _I-1, Wf wherein _I-1Be the filter word length in i-1 stage, W ₀=1, Wf ₀=0.The extraction factor M in i stage _iBe easy to draw by the down-sampling multiple, suppose the realization of FIR filter is realized with the leggy form that all therefore adopt the structure that extracts filtering more earlier, the position of extracting the factor should suitably move forward.According to above-mentioned data, can calculate the power consumption of four kinds of modes, classify following table as:

Table 2: power consumption

	P (power consumption)	Relative power consumption
			CIC+FIR 4:1	1731	100％
CIC+FIR 2:1	938	54％
			CIC+CCOS1+FIR 2:1	1539	89％
CIC+CCOS2+FIR 4:1	2356	136％

The power consumption minimum of CIC+FIR 2:1 mode, but noise power is bigger.The power consumption of CIC+CCOS1+FIR 2:1 mode is less than CIC+FIR 4:1 mode, but its noise power much smaller than CIC+FIR 4:1 mode.The power consumption maximum of CIC+CCOS2+FIR 4:1, but noise power CIC+CCOS1+FIR2:1 mode quite, and the passband of this mode is the most smooth.

The above only is preferred embodiment of the present invention, not in order to restriction the present invention, all any modifications of being done within the spirit and principles in the present invention, is equal to and replaces and improvement etc., all should be included within protection scope of the present invention.

Claims (8)

1, a kind of decimation filter of digital, the cic filter that comprises the cosine prefilter of being with cascade, it is characterized in that: also comprise compensating filter and FIR filter, the cic filter of the cosine prefilter of described band cascade, compensating filter and FIR filter link to each other successively.

2, decimation filter of digital as claimed in claim 1 is characterized in that, described compensating filter generates compensating filter by " frequency sampling method ".

3, decimation filter of digital as claimed in claim 1 or 2 is characterized in that, the inverse of the frequency response of described compensating filter for being rung by the width of cloth of the filter that compensated.

4, decimation filter of digital as claimed in claim 1 or 2 is characterized in that, described compensating filter is the compensating filter of 30 rank, 16 gains.

5, decimation filter of digital as claimed in claim 1 is characterized in that, the cic filter of the cosine prefilter of the band cascade of described decimation filter of digital comprises the cosine prefilter of cic filter and band cascade.

6, decimation filter of digital as claimed in claim 5 is characterized in that, described cascade cosine prefilter z variation is: $H_{\mathrm{CCOS}}\left(z\right)=\underset{i=1}{\overset{K}{\mathrm{\&Pi;}}}{H}_{\mathrm{COS},i}\left(z^{N_i}\right),$

Wherein $H_{\mathrm{COS},i}\left(z\right)=1/4(z^{N_i}+z^{-N_i})+1/8(z^{{2N}_i}+z^{-2N_i})+1/4.$

7, the described decimation filter of digital of claim 5 is characterized in that, described cascade cosine prefilter z variation is:

$H\left(z\right)=H_{\mathrm{CIC}}^4\left(z\right)H_{\mathrm{CIC}}^4\left(z^2\right)H_{\mathrm{CIC}}^4\left(z^4\right)H_{\mathrm{CIC}}^4\left(z^8\right)H_{\mathrm{CIC}}^4\left(z^{16}\right)H_{\mathrm{CCOS}}\left(z\right)$

H wherein _CIC(z)=1/2 (1+z ^-1),

$H_{\mathrm{CCOS}}\left(z\right)=H_{\mathrm{COS}}^{n_1}\left(z^8\right){H_{\mathrm{COS}}^{n_2}\left(z^4\right)H}_{\mathrm{COS}}^{n_3}\left(z^2\right)H_{\mathrm{COS}}^{n_4}\left(z\right),n_1=n_2=n_3=n_4=4,$

H _COS(z)＝1/4(z+z ^-1)+1/8(z ²+z ^-2)+1/4。

8, the described decimation filter of digital of claim 5 is characterized in that, described cascade cosine prefilter z variation is:

$H\left(z\right)=H_{\mathrm{CIC}}^4\left(z\right)H_{\mathrm{CIC}}^4\left(z^2\right)H_{\mathrm{CIC}}^4\left(z^4\right)H_{\mathrm{CIC}}^4\left(z^8\right)H_{\mathrm{CCOS}}\left(z\right)$

H wherein _CIC(z)=1/2 (1+z ^-1), $H_{\mathrm{CCOS}}\left(z\right)=H_{\mathrm{COS}}^{n_1}\left(z^4\right)H_{\mathrm{COS}}^{n_2}\left(z^2\right)H_{\mathrm{COS}}^{n_3}\left(z\right),n_1=n_2=n_3=4,$

H _COS(z)＝1/4(z+z ^-1)+1/8(z ²+z ^-2)+1/4。

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