INFLUENCE OF QUANTIZATION NOISE ON DFT BASED DSP ALGORITHMS

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1 XVIII IMEKO WORD COGRESS Metoogy fo Sustinbe Deveopment Septembe, 7, 6, Rio de Jneio, Bzi IFUECE OF QUATIZATIO OISE O DFT BASED DSP AGORITHMS Mtin ovotný, Mioš Sedáček Czech Technic Univesity in Pgue, Pgue, Czech Repubic, {SedceM, ovotnm5}@fe.cvut.cz Abstct: This ppe des with nysis of infuence of quntiztion noise on estimtion of pmetes of sign gined by vious goithms bsed on Discete Fouie Tnsfom (DFT) fo the cse of coheent nd non coheent smping. Theoetic nysis, numeic simution esuts nd expeiment vidtion e pesented. The nyzed methods e comped fom the point of view of thei sensitivity to quntiztion noise. Keywods: DFT, uncetinty nysis, quntiztion noise.. ITRODUCTIO A digit mesuing instuments nd systems wok with digitized signs, i.e. signs discete both in time nd in mgnitude. Smping nog sign in time esuts in peiodicity of sign fequency spectum, sign mpitude quntiztion eds to coesponding component in uncetinty of mesued sign. Quntiztion noise is dditive noise of input sign smpes nd is usuy supposed to be unifomy distibuted white noise, with stndd devition diecty popotion to the quntiztion step of the used quntize (nog-to-digit convete). Its men vue is zeo in most cses (if ounding is used in A/D convesion). Since A/D convesion is pesent in digit mesuing instuments, quntiztion noise is wys pesent in sign smped to be pocessed by the DSP goithm impemented in given instument. Tht is why nysis of its infuence to mesued vue uncetinty is vey impotnt tsk in mesuement. Mny ppes wee theefoe devoted to nysis of infuence of vious components of the digit mesuing instument stuctue on mesuement uncetinty. The smping convete ws nyzed in [], infuence of ADC quntiztion, smping jitte nd micopocesso finite wodength in bsic bsic FFT goithms fo coheent smping nd without sign windowing ws nyzed in [], [3], numeic method bsed on compute simution ws seected to nyze mesuement of men sque vue of input sign in [4], [5]. Spectum nysis of digitized sign beongs to the most common tsks in digit sign pocessing. The DFT goithm epesents the bsic DSP goithm fo the spect nysis. Since the mesued sign is most fequenty smped non-coheenty, windows e used to suppess the ekge. Vious goithms cn be used to obtin estimte of sign pmetes fom the DFT spectum of windowed sign. This ppe is focused on uncetinty nysis of some of these goithms. Quntiztion noise is supposed to be the min souce of uncetinty. Sign pmete estimtes e ffected besides quntiztion so by ekge. The infuence of ekge on estimted pmetes ws discussed in [6].. DFT AD QUATIZATIO OISE Since the estimte of sign pmetes is bsed on pocessing of DFT spectum, uncetinty of DFT spectum infuences the esutnt uncetinty of the estimte. Fo the cse of non-coheent smping, cosine windows e most fequenty ppied to the smped sign. Cosine time window w cos (n) is defined s w cos π ( n) cos n, n,,..., () whee is the window ode, is window ength nd e coefficients specific fo ech window. Fequency spectum of windowed sign cn be computed using the DFT goithm nd is given by n πn k X ( k) x( n) w( n) e () whee is DFT ength nd x(n) e input sign smpes. The smpes e ffected by quntiztion noise with stndd devition u n V EOB whee V is the fu sce nge of the used ADC nd EOB is its effective numbe of bits. This quntiztion noise cuses uncetinty of modue u M of DFT spect ine M(k) X(k) [7], um un nnpg (4) whee nnpg is nomised noise powe gin defined s [8] nnpg n ( w( n) ) Phse uncetinty cn be expessed s [7], (3) (5) un uϕ ( k ) nnpg (6)

2 3. METHODS OF PARAMETERS ESTIMATIO Vious methods e used to obtin sign pmetes of muti fequency signs (fequency, mpitude nd phse of individu hmonic o intehmonic components) fom DFT spectum in coheent nd in non-coheent smping. Sinusoid input sign is supposed in the nysis pesented hee. Resuts of this nysis e vid fo ech sinusoid component of muti sinusoid sign (sign composed of sum of sinusoid components). 3.. Coheent smping Windowing is not used in the cse of coheent smping of muti hmonic sign. Ampitude of spect component V i of the mesued sign in this cse is diecty popotion to the modue of the coesponding DFT spect ine V i k V M () k whee M(k) is the modue of DFT spect ine epesenting the i-th hmonic component nd M() epesents the spect ine coesponding to the men vue of sign V. The uncetinty due to quntiztion noise of these pmetes cn be found fom (7) u n u ( Vi ) i (8) un u ( V ) (9) The uncetinty of estimte of phse spectum ines cn be computed by (6) fo nnpg (ectngu window). 3.. on coheent smping In cse of pocessing of non coheenty smped sign (non-intege numbe of sign peiods smped), windowing is used to educe the ekge effect. Vious goithms cn be used fo detemining sign pmetes fom the DFT spectum of windowed sign. Two of them e mentioned in the next sections. 3.. RMS vue estimtion bsed on sign enegy in the window min obe Estimte of RMS vue of hmonic spectum component cn be found [8] s X RMS f f M ( f k ) + M ( f nnpg fk f fk f ) k () Hee f nd f e fequencies defining the min obe of the window spectum shifted to the sign hmonic component nd contining + DFT mpitude spectum ines. M(f k ) e components of mpitude spectum nd nnpg is nomized noise powe gin (5). Uncetinty nysis of this goithm is compicted becuse of nonzeo covince between spect ines in the min window obe. The nysis incuding this covince ws pesented in [7] nd the esutnt fomu fo uncetinty of RMS vue is un ux RMS EBW () whee EBW is equivent-noise bndwidth of the squed window (equ to fo ectngu window) nd given by [9] n EBW () n 4 w ( n) w ( n) 3.. Windowing nd intepotion in fequency domin Anothe method is bsed on sign windowing nd intepotion in fequency domin (W/IFD). Intepotion bsed on Rife Vincent windows of the fist css tht we nyze beow is descibed in []. This method is bsed on finding the exct fequency of ech spect component of the mesued sign nd on known window spectum shpe. Knowedge of fequency (in cse of non-coheent smping not pced on DFT gid) ows subsequent estimte of mpitude nd phse of the investigted spect component. The uncetinty nysis focused on the impotnt speci cse of using Hnn window is pesented in []. Estimte of exct fequency In the intepotion in fequency domin bsed on Rife Vincent windows of the fist css (RV), finding the exct fequency is bsed on tios of the oc mximum vue M(k) to its neighboing vues of the mpitude spectum M ( k + ) M ( k ) β, β (3) whee M(k), M(k+) nd M(k ) e modues of the DFT spectum of sign. Estimtes of fction fequency δ etive to fequency bin (see Fig.) coesponding to tios β nd β e given by ( + ) β ( + ) β δ, δ (4) + β + β whee is ode of cosine window (see ()). RV windows e subset of cosine windows fo the concete coefficients vues, see Tb. 5). Fig. Pincipe of intepotion in fequency domin

3 Resutnt estimte of fequency dispcement (fction fequency) δ is detemined s vege of δ nd δ δ + δ δ (5) Estimte of fequency of coesponding hmonic component is theefoe f ( k + ) f δ (6) whee f is fequency bin ( f f s /), f s is smping fequency nd is ength of DFT. Estimte of mpitude nd phse Estimte of mpitude nd phse of spect component is bsed on intepotion of sign DFT spectum to find the ppoximte vue of oc mximum of Discete-Time Fouie Tnsfom (DTFT) spectum. The DTFT spectum is continuous function of fequency, see Fig.. Spectum of the used window hs to be known. Spectum of cosine window () is η jπ η W ( η) e sin( πη) (7) π η whee η is e numbe fom the intev <, >, is ength of the DFT nd e window coefficients. Sign windowing in the time domin coesponds to the convoution of sign nd window spect in the fequency domin. Since sinusoid sign FT spectum is Dic puse pced on sign fequency, the bove-mentioned convoution mens ony shifting window spectum to the sign fequency. An estimte of mpitude of the investigted spect component cn be found fo the known δ s π A (8) D whee nd D γ ( δ ) δ sin( πδ ) (9) γ ( δ ) () δ The estimte of phse is bsed on the ineity of phse of cosine windows. Fo the known δ it cn be obtined s π ϕ nge( X ( k)) π δ + sign( D) () whee nge(x(k)) is phse of the DFT spectum (). This expession epesents the phse eted to the beginning of smping. If we e inteested in phse diffeence ony, this diffeence cn be found s the diffeence of phse vues of the windowed DFT spect (spect ekge hs the sme effect on phse vues of both signs hving the sme fequency). Uncetinty nysis Since the intepoted DFT spectum is ffected by quntiztion noise, sign pmetes bsed on this spectum e s we coupted by quntiztion noise. Estimte of uncetinty due to quntiztion cn be deived by using the w of popgtion of uncetinties []. Uncetinty of fequency A estimtes of pmetes depend on fction fequency bin δ. Theefoe so uncetinty of estimtes of these pmetes depend on uncetinty of δ. Tht is why it is necessy to find fisty the uncetinty of fction fequency bin δ. As cn be seen fom (3) (5), δ depends on some tios of modues of DFT. We sh theefoe fisty find uncetinty of tios β nd β. Fom the w of popgtion of uncetinties thee is um u ( β i ) ( β i + β i ) () whee i,, u M is uncetinty of the DFT modue (4) nd is coetion coefficient between neighboing two modues of DFT spectum (M(j) nd M(j+)). The cn be expessed s [7] + n nnpg n n+ (3) Knowing uncetinty of β nd β, it is possibe to deive uncetinty of δ nd δ u u + δ) u ( β ) ( β) (4) + ( + δ ) u ( β ) ( β ) (5) + ( Hee is the cosine window ode (). To obtin esutnt fomu fo uncetinty of δ it is necessy tke into ccount the covince between δ nd δ. These vues e dependent on β nd β tht e detemined by DFT modues (3). Since we know the covince of these modues, it is convenient fisty to deive the coetion coefficient between these quntities ( β, β ) β β ( β + β ) + ( β + β )( β + β ) (6) Hee is coetion coefficient between two modues of DFT spectum M(j-) nd M(j+). It cn be expessed s [7] + + nn+ 4 n (7) nnpg It cn be shown tht the demnded coetion coefficient between δ nd δ is δ, δ ) ( β, ) (8) ( β

4 Fig. Uncetinty of fction fequency bin δ s function of nomized fequency, Hnn window used Bsed on pevious fomue the esutnt uncetinty of δ cn be expesses s u ( δ) u ( δ) + u ( δ ) + u( δ) u( δ ) ( δ, δ ) (9) Fig. shows simution esuts nd theoetic vues of stndd uncetinty of δ due to quntiztion (sinusoid sign, V pp 9.9 V, zeo men vue, DFT ength 5, ADC fu-sce nge V V nd its effective numbe of bits EOB., epetitions, not incuding bis of δ cused by sideobes of spectum coesponding to mioed spect component). Quntiztion noise ws modeed by unifomy distibuted dditive noise. The uncetinty of fequency estimtion of sign is ccoding to (6) u( f ) u( δ ) f (3) Uncetinty of sign component mpitude Uncetinty of mpitude of sinusoid component of sign cn be found by ppying the w of uncetinty popgtion on eqution (8). We get π u ( A) A u M ( k) A A + u M ( k) δ M A + δ (, δ ) u ( δ ) + (3) whee u(m(k),δ) is the covince between M(k) nd δ nd it cn be expessed s u ( ) M ( + ) β β u, δ (3) ( + β ) ( + β) Afte expessing deivtives in (), the esutnt uncetinty of mpitude cn be witten s u ( A) + C D π A u u ( δ ) C D M + u(, δ ) (3) Fig.3 Uncetinty of sign mpitude estimtion s function of nomized fequency, Hnn window used whee D D C + δ sin( πδ ) δ cos( πδ ) nd D is defined by the eqution (9). ( δ ) (33) Dependence of uncetinty of mpitude estimtion on nomized fequency is depicted in Fig. 3. Uncetinty of phse Uncetinty of phse is ccoding to () detemined by uncetinty of DFT phse spectum (6) nd uncetinty of δ (9) u ( ϕ) uϕ( k ) + π u ( δ) (34) whee u φ(k) is uncetinty of the DFT phse spectum (6) nd u(δ) is uncetinty of fction bin δ (9). The fction fequency bin δ depends on the modue of DFT spectum. The mutu covince between fction bin δ nd the phse spectum ws expeimenty found to be negigibe. Dependence of uncetinty of phse estimtion on nomized fequency is depicted in Fig. 4. Fig.4 Uncetinty of sign phse estimtion s function of nomized fequency, Hnn window used

5 4. COMPARISO OF THE DESCRIBED METHODS Using windows eds to the ekge eduction, howeve it infuences so the uncetinty of esutnt pmete due to quntiztion. Theoetic vues of uncetinty wee veified by mesuements. Tb. to Tb. 4 ow compison of pesented theoy nd mesuements. The HP345A geneto ws used fo genetion of sinusoid with defined mgnitude nd fequency. Sign ws digitised by ow-cost mutipexed -bit ks/s DAQ pug-in bod (63E of tion Instuments). Instuments wee contoed by GPIB, MATAB tooboxes Dt Acquisition nd Instument Conto wee used. Mesuement conditions wee sinusoid sign, V pp 9.9 V, zeo men vue, DFT ength 5, smping fequency khz, ADC fu-sce nge V V nd its effective numbe of bits EOB.. Uncetinty ws evuted s stndd devition of mesuements, bis of δ cused by sideobes of spectum coesponding to mioed spect component ws not tken into ccount, becuse it is not cused by quntiztion noise. Sign fequency ws seected so tht k+δ 5.4 (6 fo the cse of coheent smping). It ws found tht fo the cse of noncoheent smping, uncetinty of the RMS vue by using ectngu window cn incese moe thn -times. When using othe windows, the chnges of uncetinty with δ is ess thn %. Tbe. Stndd uncetinty of RMS vue in cse of coheent smping (p 3.). Stndd uncetinty (V) theoetic mesued Rectngu (no window) Tbe. Stndd uncetinty of RMS vue computed fom window spectum min obe (p 3..). Window ode Stndd uncetinty (V) theo. mes. Hnn Hmming Bckmn Tems Bckmn His Tbe 3. Stndd uncetinty of RMS vue computed by mens of W/IFD method (p 3..). Window ode Stndd uncetinty (V) theoetic mesued RV, css (Hnn) RV, css RV, css RV, css Tbe 4. Stndd uncetinty of fction fequency bin δ computed by mens of W/IFD method (p 3..). Window ode Stndd uncetinty (-) theoetic mesued RV, css (Hnn) RV, css RV, css RV, css Tbe 5. Coefficients of windows Rife-Vincent, css, fo window ode Rife Vincent window coefficients / / /8 4/8 / /3 5/3 6/3 / /8 56/8 8/8 8/8 /8 Tb. 5 pesents vues of coefficients of windows Rife- Vincent, css, fo window odes up to fou. Thei vues e tken ove fom [] but chnged so tht the mximum vue of the window is equ to. 5. COCUSIO Windowing suppesses ekge on the one hnd, but inceses uncetinty of estimted pmete due to quntiztion (comped with coheent smping) on the othe hnd. The highe ode of window, the highe ekge suppession, but so the highe uncetinty due to quntiztion. Howeve, s cn be seen fom the tbes, the uncetinty inceses etivey sighty with window ode incesing. Uncetinties nyzed in this ppe epesent the uncetinty component due to quntiztion noise ony. The uncetinty component cused by ong-nge ekge fom othe sign components (e.g. the mioed component in cse of sinusoid input sign) wee mentioned in [6]. ACKOWEDGMET This esech ws suppoted by the esech pogm o. MSM "Resech of Methods nd Systems fo Mesuement of Physic Quntities nd Mesued Dt Pocessing " of the CTU in Pgue sponsoed by the Ministy of Eduction, Youth nd Spots of the Czech Repubic, nd by the esech pogm o. /5/H3 "Resech, deveopment nd optimiztion of mesuing systems nd mesuement uncetinty estimtion by thei ppiction" of the CTU in Pgue, sponsoed by Gnt Agency of the Czech Repubic.

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