Analytical Derivation of Quantization Error in Threshold Level Quantizers Using Bipolar PFM
Uniform quantization is a topic that has been extensively studied. However and although an analytical description of quantization noise has been proposed, most descriptions of the spectral properties of quantization error resort to statistical descri…
Authors: Ricardo Carrero, Ruben Garvi, Luis Hern
Analytical Deriv ation of Quan tization Error in Threshold Lev el Quan tizers Using Bip olar PFM 1 st Ricardo Carrero Electronic T ec hnology Dep. Carlos I I I Univ ersit y Madrid, Spain rcarrero@pa.uc3m.es 2 nd R ub en Garvi Electronic T ec hnology Dep. Carlos I I I Universit y Madrid, Spain rgarvi@ing.uc3m.es 3 rd Luis Hernandez Electronic T ec hnology Dep. Carlos I I I Universit y Madrid, Spain luish@ing.uc3m.es Abstract—Uniform quan tization is a topic that has b een extensiv ely studied. Ho w ev er and although an analytical de- scription of quantization noise has been proposed, most de- scriptions of the sp ectral prop erties of quantization error resort to statistical descriptions. In this pap er, w e show that the sp ectrum of a quan tized signal can b e expressed using pulse frequency mo dulation. W e rst establish the equiv alence of a uniform quantizer with a system based on the bipolar pulse frequency mo dulation and we dene afterw ards the F ourier transform of the quantized signal using pulse frequency mo dulation prop erties. This mo del brings a more intuitiv e understanding of the spectral structure of quantization noise and complements prior research in the topic. The results of the pap er can be directly applied to level crossing ADCs with zero-order-hold in terp olators, giving an accurate estimation of their p erformance. Index T erms—quan tization, sigma-delta modulation, pulse- frequency mo dulation, lev el crossing ADC I. Introduction Analysis of quantization noise in uniform quantizers is a topic addressed ov er the past decades in several w orks, as for instance [1]–[6]. In [1] a closed form for the quantization noise of an arbitrary con tinuous-time signal is provided. Ho wev er, most papers use a statistical approac h to compute the sp ectrum of quantized and sampled signals. The Lev el crossing ADC is an architec- ture [7] suitable for lo w p ow er edge pro cessing of time- sparse signals [8]. Although Signal to Quantization Noise Ratio (SQNR) of lev el crossing ADCs can b e impro ved b y sophisticated reconstruction algorithms [7], zero-order- hold in terp olation is frequently used due to its simplicity [9]. In this case, level crossing ADCs can b e mo deled as unsampled uniform quantizers, which motiv ates the in terest in its analytical description. In tegral Pulse F requency Modulation (PFM) has been link ed to sigma-delta mo dulation [10]. In this pap er, we in v estigate the possibility of a similar equiv alence sc heme b et w een PFM and uniform quantization. W e analyze an op en lo op uniform quantizer, showing that threshold quan tization resorts to a pulse frequency modulation as w ell, although referred to the deriv ative of the input signal. Research funded by grant PDC2023-145850-100 of the Spanish Ministry of Science, Innov ation and Universities With this PFM model w e will calculate a series expansion of the output of a uniform quan tizer for a sin usoidal input. This series shows that the quantization error sp ectrum is composed of an innite num b er of input harmonics organized in mo dulation sidebands. The amplitudes of the input harmonics in the modulation sidebands are describ ed by a series of Bessel functions cen tered around DC. As a fundamen tal dierence to [10], use of bipolar pulse frequency mo dulation is comp elling in the analysis of a uniform quan tizer and is the reason why mo dulation sidebands appear all cen tered at DC. The pap er rst establishes a PFM-based model of a uniform quantizer, pro ving the need for a bipolar PFM for the model to b e correct. Afterwards, the mathematical denition of bipolar PFM is laid out, and the existing mathematical description of unip olar PFM is extended to bip olar PFM. Finally w e v alidate our mo del confronting sim ulation cases with theoretical predictions. The results of this paper allow to mo del accurately the operation of a level crossing ADC with zero-order-hold in terp olator. I I. PFM equiv alent of a uniform quan tizer As pointed out in the introduction, w e ha ve to distin- guish betw een threshold lev el quan tization and sampling. W e are going to refer here to threshold level quantization but without considering a sampler afterwards, as a dier- ence to [3]. As start point, we will show that the in tegral of a PFM enco ded signal is iden tical to the quantized in tegral of said signal. A. PFM Enco ded Integrator Fig. 1.a shows an integrator that in tegrates signal x ( t ) > 0 . The output, v ( t ) , is then quan tized with a uniform quan tizer with quantization step ∆ , pro ducing signal y q ( t ) . Fig. 1.b, sho ws a PFM encoder that enco des the same signal x ( t ) with PFM modulation [11]. In [10], [12] the t ypical system emplo y ed to implement a PFM mo dulator is describ ed as a feedback lo op in volving integration and a threshold detector. The output of a PFM mo dulator consist of a train of Dirac delta impulses d ( t ) at instan ts t k dened b y the equation [12]: ∫ x(t) v(t) y q (t) ∫ x(t) PFM d(t) y p (t) a) b) Fig. 1. a) Integrator follow ed by uniform quan tizer. b) PFM equiv alent of a) d ( t ) = ∞ ∑ k =0 ∆ · δ ( t − t k ) t k | ∫ t k t k − 1 x ( t ) dt = ∆ (1) W e hav e a Dirac delta impulse in d ( t ) ev ery time the in tegral of x ( t ) increases b y an amoun t ∆ . The main prop ert y of PFM modulation is that the av erage v alue of the Dirac delta impulses in d ( t ) follo ws the input signal, b eing a simple wa y to enco de a lo w-pass con tinuous signal in to a pulsed one. Observing the denition in (1), it is clear that we hav e a delta impulse in d ( t ) of Fig. 1.b ev ery time signal v ( t ) in Fig. 1.a crosses a threshold of the uniform quantizer. No w we in tegrate d ( t ) whic h turns the summation of delta impulses into a summation of step functions y p ( t ) : y p ( t ) = ∆ · ∞ ∑ k =0 u ( t − t k ) (2) where u ( t ) represents the Heaviside step function. As can be seen, signals y p ( t ) and y q ( t ) are mathematically iden tical and hence, the systems of Fig. 1.a and Fig. 1.b can b e considered equiv alent. Note that this equiv alence do es not stem from the low pass signal approximation prop ert y of PFM modulation but from a mathematical iden tit y . B. Dierential-In tegral model of a threshold level quan- tizer The mo del of Fig. 1 shows the connection b et w een quan- tization and PFM but requires an integration. T o mo del a con ven tional uniform quantizer, w e can complement the mo del using a dierentiation function cascaded with the in tegration, as sho wn in Fig.2.a. The integration function ∫ y q (t) ∫ x(t) x(t) PFM d(t) y p (t) a) b) dt d dt d ω (t) Fig. 2. a) Uniform quantizer. b) Dieren tial-In tegral mo del. in fron t of the quan tizer would b e comp ensated by the dieren tiation blo ck. This setup, can b e repro duced in the PFM equiv alent by placing a dieren tiator in fron t of the PFM enco der, which no w enco des signal w ( t ) , the deriv ativ e of x ( t ) . F or simplicity , we will assume x ( t ) to b e zero mean in the foregoing. This conguration has ho w ev er, some inconsistencies regarding the denition of the PFM mo dulator which need to b e solved. Regardless of the input signal DC level, the DC conten t of signal w ( t ) will alwa ys b e zero due to the deriv ativ e. Hence, signal w ( t ) will ha v e p ositive and negative v alues depending of the slop e of x ( t ) , but the denition in (1) requires the PFM input to b e alw a ys positive. As a consequence, we need to mo dify the denition of the PFM signal to represent b oth p ositive and negativ e signals. Note that adding a DC lev el to w ( t ) w ould not solve the problem as the threshold crossing points of y q ( t ) w ould no longer be coincident with those of y p ( t ) . A feasible solution instead is to redene the PFM mo dulation as follows: d ( t ) = ∞ ∑ k =0 P k · δ ( t − t k ) P k = ∫ t k t k − 1 x ( t ) dt t k | | P k | = ∆ (3) where |·| represen ts the absolute v alue. This denition of the pulse frequency modulation will be referred as Bip olar PFM and retains the prop erty of enco ding the lo w pass con ten ts of the input signal in the a v erage v alue of the input signal. Fig. 3 sho ws a sim ulation of the system of Fig. 2 using an input sinusoid x ( t ) with an amplitude A=5, frequency f x = 2 mH z (see Fig. 3.a) and a quantizer with quan tization step ∆ = 1 , whic h produces signal y q ( t ) , plotted in Fig. 3.e. After deriv ation, signal v ( t ) is plotted in Fig. 3.b. The bipolar PFM signal, d ( t ) , is represen ted in Fig. 3.c, where we can see that delta impulses are coinciden t with the transitions betw een quan tization levels t 0 100 200 300 400 500 600 700 800 900 1000 x(t) -5 0 5 a) t 0 100 200 300 400 500 600 700 800 900 1000 v(t) -0.05 0 0.05 b) t 0 100 200 300 400 500 600 700 800 900 1000 d(t) -100 0 100 c) t 0 100 200 300 400 500 600 700 800 900 1000 y p (t) -5 0 5 d) t 0 100 200 300 400 500 600 700 800 900 1000 y q (t) -5 0 5 e) Fig. 3. a) Input sin usoid. b) Input deriv ative. c) bipolar PFM. d) Signal y p ( t ) . e) Signal y q ( t ) of y q ( t ) . Finally , after integration of d ( t ) , we obtain y p ( t ) , plotted in Fig. 3.d, which is equal to y q ( t ) . Applying a DC input to the dierential-in tegral mo del is possible but requires some further explanation. If w e apply to a uniform quantizer a signal whose DC con ten t is not zero, there will be also a quantized DC oset at the output. Giv en the input deriv ative in Fig. 2.b one could think that no DC input can be encoded in our mo del. Ho w ev er, if we assume that the input is causal and there is a step at t = 0 when the DC v alue is applied, there will b e an input impulse at the deriv ative output which will set the prop er initial v alue in the PFM mo dulator internal in tegrator translating in an initial in teger num b er of delta pulses in d ( t ) accoun ting for the right DC quantized v alue. I I I. Sp ectral Analysis of PFM Quantizer Mo del The bip olar PFM quan tizer mo del describ ed in Fig. 2 do es not represent more than just an alternative p oint of view of quantization. Ho wev er, this representation allo ws to use the well kno wn sp ectral analysis of PFM signals to study the sp ectral structure of quantization error [2], [4] in a more intuitiv e wa y , also providing analytical results. A series expansion for PFM signals expressed as Dirac delta pulses enco ding a sin usoid can b e found for instance in [12]. This expansion can b e extended to an arbitrary n um ber of sin usoids [13] enabling a broader signal con text. Let’s assume that an input sinusoid v ( t ) with DC v alue v m , amplitude B and frequency f x , is applied to a PFM mo dulator as dened in (1), we can write: Fig. 4. a) Single p olarity PFM b) Bip olar PFM v ( t ) = v m + B · cos(2 π f x t ) (4) Then, signal d ( t ) can b e expanded in to a trigonometric series as follo ws: d ( t ) = f 0 + B · cos(2 π f x t ) + m ( t ) , m ( t ) = 2 f 0 · ∆ · ∞ ∑ q =1 ∞ ∑ r = −∞ J r ( q · B f x · ∆ ) · ( 1 + r f x q f 0 ) cos(2 π ( q f 0 + r f x ) t ) (5) where f 0 corresp onds to the rest frequency of the PFM mo dulator, which is the frequency produced when the DC v alue v m is applied, and J r ( · ) are the Bessel functions of the rst kind. This analysis shows that a PFM signal can b e decomp osed into t wo comp onents, one represen ting the input signal itself and another composed of modulation sidebands m ( t ) . The modulation sidebands are groups of tones shifted b y in teger multiplies of f x around rest frequency f 0 and its harmonics. This situation is depicted in the example sp ectrum of Fig. 4.a. In the system of Fig. 2, the signal has zero mean b y denition and then, v m and f 0 are zero. Note that in [12], no restriction is placed in the v alues of v m and f 0 for the pro of of (5) to b e v alid, hence it applies to Bip olar PFM as w ell. Then, w e can write: x ( t ) = A · sin (2 π f x t ) v ( t ) = d ( x ( t )) dt = B · cos (2 π f x t ) B = 2 π f x · A d ( t ) = v ( t ) + m ( t ) y p ( t ) = x ( t ) + ∫ t 0 m ( τ ) dτ (6) Applying (6) to (5) we obtain: t 0 100 200 300 400 500 600 700 800 900 1000 d(t) -2 -1 0 1 2 a) t 0 100 200 300 400 500 600 700 800 900 1000 d a (t) -2 -1 0 1 2 b) t 0 100 200 300 400 500 600 700 800 900 1000 y pa (t) -5 0 5 c) t 0 100 200 300 400 500 600 700 800 900 1000 y q (t) -5 0 5 d) Fig. 5. a) simulated PFM. b) Analytical PFM. c) Analytical quantized signal. d) Sim ulated quantized signal. d ( t ) = B · cos(2 π f x t ) + m ( t ) , m ( t ) = 2∆ · ∞ ∑ q =1 ∞ ∑ r = −∞ J r ( q · B f x · ∆ ) · ( r f x q ) cos(2 π r f x t ) (7) In (7) we can see that m ( t ) is comp osed b y harmonics of the input frequency f x only , weigh ted by the Bessel functions and indexes q and r. This situation is depicted in the example of Fig. 4.b. The quantized signal can b e obtained by integration. W e can decompose the quantized signal y p ( t ) as the sum of the input signal x ( t ) and the quan tization error e ( t ) : y p ( t ) = A · sin(2 π f x t ) + e ( t ) , e ( t ) = ∆ · ∞ ∑ q =1 ∞ ∑ r = −∞ J r ( q · 2 π · A ∆ ) · ( 1 π q ) sin(2 π r f x t ) (8) Since (8) consists of a sum of sine wa ves, we can analytically compute its F ourier transform, a result sho wn in (9): Y p ( w ) = A · π · ( δ ( w + w x ) − δ ( w − w x )) j + E ( w ) , E ( w ) = ∆ · ∞ ∑ q =1 ∞ ∑ r = −∞ J r ( q · 2 π · A ∆ ) · 1 q · ( δ ( w + r w x ) − δ ( w − r w x )) j (9) Prior works mention that quan tization noise can b e expressed as frequency mo dulated sinusoids [1] and the sp ectrum of uniformly sampled quantizers is link ed to Bessel functions [3]. How ev er, (8) applies to the con tinuous w a v eform pro duced b y an unsampled threshold quantizer and hence, is v alid for lev el crossing ADCs with zero-order- hold in terp olation. IV. Simulations T o prov e the calculations of the previous section, we ha v e compared the time domain b ehavioral sim ulation of Fig. 3 with a numerical ev aluation of (7) and (8). Fig. 5.a repro duces the PFM signal of Fig. 3.d for reference. Fig. 5.b shows the equiv alent result computed analytically using (7), whic h displa ys similar delta impulses. The sim u- lation has been made truncating the innite summations, using 1000 harmonics of f x (index r) and 50 mo dulation sidebands (index q). Fig. 5.c repro duces the same data for the series in (8) which can be contrasted with the original quan tized signal, repro duced in Fig. 5.d. Due to the nite n umber of summation terms used in the series expansion, the Gibbs phenomena can be appreciated in Fig. 5.b as a ringing around the delta impulses. This eect w as also previously iden tied in [1]. As can b e seen, the PFM analytical mo del accurately represents quantization using the w ell kno wn PFM mathematical deriv ations. The series expansion of (8) can predict the sp ectral densit y of a quantized sinusoid, by representing the amplitude of each harmonic of f x . W e hav e simulated the system of Fig. 2 for a sine w av e with amplitude A=512, representing the outcome of a 10 bit quan tizer at full scale. Fig. 6.a represents the FFT of the quan tized sin usoid, suppressing the input tone. Fig. 6.b represen ts the amplitudes of the harmonics of f x , computed using (8) in dB. A similar spectrum can b e observ ed, corresponding with the prediction of Fig. 4.b. F or further insight, Fig. 6.c sho ws eac h modulation sideband spectrum, computed using (8) and display ed individually in dB, from q=1 (blue) up to q=10 (yello w). V. Conclusions Quan tization is a w ell kno w pro cess, but requires ad- v anced mathematical tools to b e describ ed, resorting to statistical calculations in some cases. In this paper w e ha v e established the connection betw een quan tization and Pulse F requency Mo dulation. This nov el p oint of view, pro vides an intuitiv e approac h to estimate the sp ectral prop erties of quantized signals, b y lev eraging the well kno wn PFM theory . In addition, we ha v e derived an equation to describe analytically the quan tization error of sin usoidal wa veforms. This result can b e directly applied to level crossing ADCs using zero-order-hold interpolation. f (Hz) 0 5 10 15 20 25 30 Y q (dB) -100 -80 -60 -40 -20 0 a) f (Hz) 0 5 10 15 20 25 30 Y pa (dB) -100 -80 -60 -40 -20 0 b) f (Hz) 0 5 10 15 20 25 30 modulation sidebands -100 -80 -60 -40 -20 0 c) Fig. 6. a) Sim ulated sp ectrum. b) Analytical sp ectrum. c) Sideband decomposition. References [1] C. M. Zierhofer, “Quan tization noise as superp osition of frequency-modulated sin usoids,” IEEE Signal Pro cessing Let- ters, vol. 16, no. 11, pp. 933–936, 2009. [2] R. Gra y and D. Neuho, “Quantization,” IEEE T ransactions on Information Theory , v ol. 44, no. 6, pp. 2325–2383, 1998. [3] R. 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