A Design Procedure for Stable High Order, High Performance Sigma ...

A Design Procedure for Stable High Order, High Performance Sigma-Delta Modulator Loopfilters Georgi Tsenov and Valeri Mladenov Dept. of Theoretical Electrical Engineering, Technical University of Sofia, 8, Kliment Ohridski St., Sofia 1000, Bulgaria {gogotzenov,valerim}@tu-sofia.bg

Abstract. In this paper we present the ideas for design of stable high order single bit sigma-delta modulator loopfilter transfer functions that provide high signal-to-noise ratio. The procedure is backed up with example and results made for third order loopfilter sigma-delta modulators and give the performance impact on them when varying the poles of the noise transfer function, when using optimized zeroes. This loopfilter function design is computed with fast theoretical calculation of the signal to noise ratio with mathematical formula, instead of approximation based on simulations and combined with theory that presents approximated value for modulator’s maximal stable DC input signal, resulting in design without the need of simulations of the modulator output bitstream. Keywords: Sigma-delta modulators, digital signal processing, stability, analogto-digital conversion, signal-to-noise ratio.

1

Introduction

From decades ago Sigma-Delta modulators are the standard for analog to digital conversion nowadays. When using high oversampling ratios sigma-delta modulators (SDM) can achieve very high signal to noise ratio (SNR) even with small number of quantization levels [1]. They shape the noise and push it to frequencies higher than the operational band of interest. Thanks to its simplicity, single bit code shaping SDM are of greatest interest, because the provided performance is influenced mostly by the loopfilter transfer function and the modulator’s oversampling ratio (OSR) and the modulator output is encoded into a bitstream. In the practice the modulator’s maximal stable DC input signal range and its SNR are determined by simulations, which also leave a zone of uncertainty. Furthermore a lot of engineers experiment with the loopfilter coefficients in order to achieve higher SNR, but up to date there is still no such a thing as optimal loopfilter transfer function for specific modulator order that provides both high performance and stable modulator behavior. Also, the realistic high performance loopflter transfer functions have the poles grouped into a complex conjugate pairs and one real pole when having odd modulator order and complex conjugate pairs for even loopfilter orders. In order to increase modulator performance

V.M. Mladenov and P.C. Ivanov (Eds.): NDES 2014, CCIS 438, pp. 114–124, 2014. © Springer International Publishing Switzerland 2014

A Design Procedure for Stable High Order

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some authors move one of the complex conjugate pair of poles [3] or the real pole [4] a little bit outside of the unit circle, while keeping the other poles inside resulting in increased SNR and reduced stability limit for maximal DC input signal amplitude beyond which the modulator becomes unstable. Moving a single pole or complex pair a little bit outside the unit circle does not necessarily makes the SDM unstable, as it is a nonlinear system, which makes the SDM behavior analysis harder for those cases. This paper is presenting a design approach for a higher odd order SDM taking into account the SDM stability and SNR performance. The approach includes variation of the zero positions on the unit circle of sigma-delta modulator loopfilter transfer function with optimized poles, with pole positions given by the delta sigma toolbox. For that type of analysis a parallel decomposition form of the loopfilter given in [2] is used, because with it there is a theory that can predict the stability range. The results presented in [2] allows approximation of the maximal stable DC input signal value without the need of simulations for single bit quantizer modulators with this particular filter form and they are used in the design procedure. For faster SNR calculation a derivation of it from the loopfilter noise transfer function is used, because in this case there is no need of modulator’s output bitstream resulting in no need of SDM simulations [5], resulting in reduced number of calculations. The paper is organized as follows. In the next chapter a theoretical background necessary for understanding the approach is given and description of the method for determination of SDM stability and the method for fast SNR approximation. Then in the third chapter is presented the design procedure targeted to obtain SDM loopfilter transfer functions providing both decent performance and guaranteed stable modulator behavior with presented the procedure results, Then in fourth chapter final conclusion remarks are given.

2

Theoretical Background

For better understanding the design approach here we will remind briefly the results in [2] that are used. The well-known basic structure of an SDM is shown in Fig.1, and consists of a filter with transfer function G(z) followed by a one-bit quantizer in a feedback loop.

Fig. 1. Basic sigma delta modulator structure

The system operates in discrete time and the input to the loop is a discrete-time sequence u(n)∈[-1, 1], appearing in quantized form at the output. The discrete-time sequence x(n) is output of the filter and quantizer input. Quantizer producing an output of +1 when its input is positive and –1 when its input is negative (single-bit), will not provide a good approximation to its input signal and for that reason an

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feedback loop is used, acting in such a way as to shift this quantization noise away from a certain frequency band. If an input signal from within this frequency band is applied to the loop, most of the noise imposed by the quantization process will be moved outside of the frequency band of interest and can subsequently be filtered out, leaving a good approximation to the input signal. This process is called noise shaping. The modulator stability can be obtained without simulations. In [2] authors consider a Nth order modulator with a loop filter transfer function of the form G( z) =

a1 z −1 + ... + aN z − N 1 + d1 z −1 + d 2 z −2 + ... + d N z − N

(1)

In the general case the loop filter transfer function have complex conjugated roots. Without loss of generality we will consider only one pair of complex conjugated roots. In this case (1) becomes G( z) =

b1 z −1 b z −1 B z −1 + B z −2 + ... + G2 ( z ) = 1 −1 + ... N −1 −1 N −2 −1 1 − λ1 z 1 − λ1 z 1 − d1 z − d 2 z

(2)

where the coefficients bi, i=1,2,…,N of the fractional components can be found easily (1 − λi z −1 ) bi = G( z) . using the well-known formula z −1 z =λ i

The denominator of the last part of (2) has a complex conjugated pair of roots and therefore (2) becomes:

G( z ) =

b1 z −1 b z −1 b z −1 + ... + N −1 −1 + N −1 −1 1 − λ1 z 1 − λ N −1 z 1 − λN z

(3)

where

λN −1 = α + jβ , λN = α − jβ bN −1 = δ − jγ , bN = δ + jγ

(4)

i.e. λN-1, λN and bN-1, bN are complex conjugated numbers. Because of this in [2] the parallel presentation given in Fig.2 of third order modulator is used. The values of the last two blocks are complex, but the output signal of these two blocks is real. They correspond to a second order SDM with complex conjugated poles of the loop filter transfer function G(z). Both signals x2 and x3 are complex conjugated, namely

x2 ( k + 1) = m(k + 1) + jn(k + 1) x3 (k + 1) = m(k + 1) − jn( k + 1)

(5)

Because of this the input of the quantizer is real i.e. (δ − jγ ) x2 ( k ) + (δ + j γ ) x3 ( k ) = 2δ m ( k ) + 2γ n ( k )

(6)


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The modulator could be considered as three first order modulators interacting only through the quantizer function. The connected signals with two modulators are complex, but the input and output signals (u and y) are the “true” signals of the modulator. As it is stressed in [2] both modulators work cooperative, because their signals are conjugated. These modulators do not exist in the real SDM and they are introduced to help the analysis of the behavior of the whole system.

Fig. 2. Block diagram of higher order SDM with parallel loopfilter form

The benefit of this modulator representation is because we can determine whenever the modulator is stable or not by this criterion [2]: (2 − λ1 )

λ1

N −2 b b1 2 | δ (1 − α ) + γβ | > − i + (λ1 − 1) (1 − α ) 2 + β 2 i = 2 λi − 1

(7)

Additionally we can also determine the maximal range of input signal ensuring the stability expressed by ∆u: 2 δ (1 − α ) + γβ b1 (2 − λ1 ) + (1 − α )2 + β 2 λ1 (λ1 − 1) i =2 i Δu < N −2 b 2 δ (1 − α ) + γβ b1 − i + λ1 − 1 i = 2 λi − 1 (1 − α )2 + β 2 N −2

bi

 λ −1 −

(8)

The fast approximation formula that will be used for fast Signal to Noise Ratio (SNR) calculations is derived in the following way in [5]. The theory of quantization and the corresponding noise is well-established. The distance between 2 successive quantization levels is called the quantization step size, Q. For a quantizer with a specified number of bits covering the range from +1 to -1 there are 2bits quantization levels and the width of each quantization step is: Q=

2 (2

bits

- 1)

(9)

The quantizer assigns each input sample u(n) to the nearest quantization level. The quantization error is simply the difference between the input and output to the quantizer, eq=y(u)-u, and is bounded by

118


-

Q Q £ eq ( n ) £ . 2 2

(10)

The quantization noise power is given by 1 σ e2 = Q

Q 2



eq 2 deq =

Q − 2

Q2 1 . = bits 12 3(2 -1)2

(11)

Many authors propose σe2 to be approximated to be 1 . (12) 3 ⋅ 22 bits This quantization error is on the order of one least-significant-bit in amplitude and it is quite small compared to full-amplitude signals. The average power of a sinusoidal signal of amplitude A, x(t)=Acos(2πt/T), is

σ e2 ≈

T

1 A2 (13) ( A cos(2π t / T ))2 dt =  2 T 0 If we assume that the signal is oversampled, then rather than acquiring the signal at the Nyquist rate, 2fB, the actual sampling rate is fs=2r+1fB, with oversampling ratio OSR=2r=fs/2fB. The quantization noise is spread over a larger frequency range yet we are still primarily concerned with the noise below the Nyquist frequency. Most of the noise power is now located outside the signal band. The quantization noise power within the band of interest has decreased by a factor OSR. The signal power occurs over the signal band only, so it remains unchanged and is given by (13). Many authors use the linear SDM model for analysis. The linear model contains two inputs: the input signal X(z) and the quantization error E(z). As the basic model shown on Fig.1 a filter is placed in front of the quantizer, known as the ‘loop filter’ and the output of quantization is fed back and subtracted from the input signal, as shown on Fig. 3.

σ x2 =

Fig. 3. Representation of a sigma-delta modulator using the linear model

This may be represented by transfer functions applied to both the input signal and the quantization noise. The Z-domain output may be represented as Y ( z ) = STF ( z ) X ( z ) + NTF ( z ) E ( z )

,

(14)

where the STF is the Signal Transfer Function and the NTF is the Noise Transfer Function. The input to the loop filter is X(z)-E(z) so that Y(z)=G(z)[X(z)-Y(z)]+E(z). Rearranging terms, we have:


STF ( z ) =

1 G( z ) , NTF ( z ) = . 1 + G( z) 1 + G( z)

119

(15)

Utilizing the linear model of the SDM the noise shaping in the SDM implies a variable noise power in the baseband: fB



σ n2 =

Se 2 ( f ) | NTF ( f ) |2 df

− fB

(16) where Se2(f)=σe2/fs is the power spectral density of the unshaped quantization noise. The total noise power, σe2, remains unchanged, but appropriate choice of the NTF(z) pushes the noise up to the high frequencies. By definition the SNR is calculated on basis of: s2 SNR(dB)=10log10 x2 (17) sn We can take the σe2, σx2 and σn2 terms to substitute them to get the general formula for the SNR of any sigma-delta modulator. SNR(dB)=10log10

s =10log10 s 2 x 2 n

A2 2 fB

òS

e

2

( f ) | NTF ( f ) |2 df

- fB

=10log10

A2 ⋅ fs fB

(18)

2se2 ò | NTF ( f ) |2 df - fB

Applying approximation of σe we get the formula: 2

SNR(dB) »10log10

3⋅ 22bits ⋅ A2 ⋅ fs fB

2 ò | NTF( f ) |2 df

fB

»10log10 3⋅ 22bits A2 fs -10log10 2 ò | NTF( f ) |2 df (19) - fB

- fB

Using numerical integration this equation can be solved for a SDM with arbitrary noise transfer function, oversampling rate and bit length. These calculations for the SNR approximation, when done with computer, are very fast and precise and are computed much faster than the SNR approximate estimation when using modulator output bitstream simulations. If using a loopfilter transfer function of odd order, using the function coefficients with Eq.8 the stability can be calculated without simulations, while with Eq.19 the SNR of the SDM can be computed without simulations. This provides the tools for SDM analysis without the need of SDM model simulations. The practical relation of ∆u and modulator stability depending on its signal value is given for one example on Figure 4 with usage of loopfilter transfer function that produces loss of stability for input signals with amplitude values lesser than 1 if 1 is the scaled maximal input signal. Usually with appliance of higher input signal the SNR rises, but here when rising the test sine wave amplitude we can observe that at some point there is an SNR decrease and eventually loss of stability. This means that the modulator can be stable for input signals with value higher than that of ∆u, but then we can’t guarantee its stability (∆u=0.68 in this example).

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Fig. 4. Relation between ∆u and the value of the input signal

3

Design Approach

Some authors [6], claims that if the NTF zeroes (loopfilter poles) are placed not on the DC, but spread in the baseband, the SNR is higher and the quantization noise is spread evenly in the baseband as shown on Figure 5.

Fig. 5. SDM noise distribution comparison with and without optimized NTF zeroes

The Delta Sigma Toolbox [7] for MATLAB, created by this author, can generate such NTFs of arbitrary order. This Toolbox was used to generate an exemplar noise transfer function with optimized zeroes for third order SDM filter order, which is: NTF(z)=

z 3 − 2.999 z 2 + 2.999 z − 1 z − 2.1992 z 2 + 1.6876 z − 0.4441 3

(20)

For this type of noise transfer function with Eq. (8) we can determine the SDM stability ranges even when moving the real pole outside the unit circle and we can find the SNR very fast with the following constrained optimization procedure:


R = max ( SNR )

(1 − zero _ ntf * z ) * (1 − zero _ ntf * z ) * (1 − zero _ ntf (1 − pole * z ) * (1 − pole * z ) * (1 − pole * z ) −1

NTF =

−1

1

2

−1

NTF ( z ) =

* z −1 )

−1

−1

2

1

3

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3

b3 z −1 b1 z −1 b2 z −1 1 ; G( z) = + + −1 −1 1 + G( z) 1 − zero _ ntf1 z 1 − zero _ ntf 2 z 1 − zero _ ntf 3 z −1

zero _ ntf 2 = α + j β , zero _ ntf 3 = α − j β , b2 = δ − jγ , b3 = δ + jγ SNR = 10log10 3 ⋅ 22 bits A2 f s − 10log10 2

fB

 | NTF ( f ) |

2

df [ dB ]

− fB N −2

Δu