Spur-Free and Stable Digital Delta-Sigma Modulator via Using HK-EFM

. This paper presents a new multi-stage noise-shaping (MASH) digital delta-sigma modulator (DDSM), denoted as HJ-MASH DDSM which can remove spurs in the output spectrum and has long sequence lengths than Jinook structure and HK-MASH in the worst input case by applying HK-EFM in the first stage error feedback modulator (EFM) of the DDSM. In addition, the proposed HJ-MASH DDSM lowers the difficulty of constructing higher order MASH DDSMs compared to HK-MASH DDSM. While the performance and stability are all superior to Jinook structure for all constant inputs, especially for the input which correspond to minimum cycle lengths. The digital delta-sigma modulator applied in fractional-N frequency synthesizers amelio-rates the synthesizer’s spectral purity apparently by verification.


Introduction
The frequency divider is one of the most significant components of frequency synthesizer, which can support frequency synthesizer to generate multiple high-precision frequency signals.According to divisor, it can be divided into integer divisor and fractional-n divisor.The fractional-n frequency divider can be constructed by using two integer dividers, a divide-by-n divider and a divide-by-(n+1) divider [1][2].
The digital delta-sigma modulator (DDSM) is extensively analyzed and used in modern wireless communication subsystems, such as fractional frequency synthesizers, analog-to-digital and digital-to-analog converters, and switched-mode power supplies to acquire an excellent performance and spectral purity.Delta-sigma modulator translate a fine input sequence that should be represented in m bits to a coarse output sequence represented in n bits, where n is significantly less than m.Delta-sigma modulator is a technique for improving the effective resolution of a quantizer by oversampling and noise shaping.
The oversampling ratio OSR is defined as: where   is the sampling frequency and   is the largest frequency component in the signal trum.2  corresponds to the Nyquist frequency.Besides, the low frequency part of the rest of its spectrum is attenuated and its high frequency part is amplified.In that case, the oversampling and the noise shaping property of the DSM can be used together to achieve a high signal-to-quantization-noise ratio using a relatively coarse quantizer [3][4].
Delta-sigma modulator (DSM) is widely used to select a clock divider in the fractional-N frequency synthesizer.As shown in Fig. 1, the conventional first-order DSM consists of a digital accumulator, a register, a quantizer and feedback loop.Since the DSM is a finite-state machine (FSM), its output sequence is periodic when the input is a constant value.More importantly, the quantization noise sequence is also periodic, and the noise sequence period or the sequence length is strongly dependent on the input value and the DSM structure.If the sequence length is short, the quantization noise power is spread over a small number of tones, resulting in spurs in the noise spectrum.
The first-order EFM is widely used in implementing the first-order DSM, since the error feedback modulator (EFM) has a relatively low hardware cost and low power consumption.We show the block diagram of a first order EFM1 that used a multi-bit quantizer.The input to the modulator is a digital word with n bit.When the v[n] is greater than M (2 n ), the quantizer overflows and the output signal y [n] (the carry out) will be one.Otherwise, the quantizer does not overflow and y[n] will be zero as follow:

Previous MASH architecture
By contrast with the SQ-DDSM and EFM, both of which use a single quantizer, the multistage noise shaping (MASH) technique allows one to realize high order noise shaping using lower modulators.In MASH DDSM, one can use lower order modulators (with orders as low as 1) in a cascade.If first order stages are available, L stages can be combined to form an L th order MASH modulator.A MASH DDSM with 1-bit internal first order modulators is a feedforward structure and is unconditionally stable; this is the principal advantage of the MASH modulator over the SQ-DDSM topology.Furthermore, in a MASH DDSM, the stable input range is equal to the full scale while the stable input range is only a fraction of the full scale in an SQ-DDSM.The MASH DDSM is widely used in commercial fractional-N frequency synthesizer products.

HK-MASH architecture
The basic building block of the HK-MASH DDSM [5] is the first order HK-EFM1 shown in Fig. 2, it is very similar to the convention EFM1.The main difference compared to the conventional EFM1 shown in Fig. 1 in that it includes a feedback block denoted as  −1 .Therefore, the cycle lengths of conventional EFMs are maximized by introducing a feedback path from the output of each EFM to its input using a delay block with a particular choice of coefficient a, where a is a small integer that is chose such that  −  is primer.
The parameter  is the normalized version  of given by

HK-EFM1 HK-EFM1 HK-EFM1 HK-EFM1
Error Cancellation Network By contrast with a conventional modulator ( = 0), a pole at  =  is added to both the STF and the NTF.If  is sufficiently small, this pole is very close to the origin in the  plane; equivalently, it is a distant pole which does not significantly affect the overall operation of the modulator.In order to illustrate this point, consider the magnitude response of the multiplying factor  ℎ () = The sequence length of the Lth HK-MASH DDSM have been proven mathematically to be equal to ( − )  .Therefore, the sequence length of the HK-MASH DDSM3 is ( − ) 3 .
Fig. 5 shows the simulated power spectrum of the HK-MASH DDSM3 when the input is half-scale, once again with zero initial condition and no dither.Note that the spectrum is close to the ideal white noise case, and no individual tones can be seen.

Jinook structure
The basic building block of the Jinook MASH structure [6][7] is the first order MEFM1 shown in Fig. 6, it is very similar to the convention EFM1.The main difference compared to the conventional EFM1 shown in Fig. 1 in that it takes two inputs and generates two ouputs to be fed to the next EFM.In the MEFM1 , the output is delivered to not only the noise cancellation network but also the subsequent MEFM1.Therefore, the Jinook MASH structure links the EFMs by connecting the output and the quantization error signal of an EFM to the input of the next-stage EFM.
x Besides, it's significant to note that the block of the first stage is a conventional EFM1, therefore, the input of EFM1 is a pure dc signal and not a perturbed one.In the MEFM1, the output and the quantization error signals that are to be connected to the next stage have the same sequence length.Since two periodic inputs can be regarded as one periodic signal, the MEFM1 is functionally equivalent to the traditional EFM, and it generates a periodic output signal.

EFM1 MEFM1 MEFM1 MEFM1
z Error Cancellation Network The output of the L th Jinook MASH DDSM can be expressed in the Z-domain as:  Although the noise shaping effect of HK-MASH is amazing, it is complicated to construct the hardware implementation of HK-MASH and the input is not exactly equal to the output.Besides, the noise shaping effect of the Jinook structure exhibits instabilities in some input cases and the minimum output sequence length is dependent on the output sequence length of the first-stage EFM cell.In this section, we proposed a new MASH structure shown in Fig. 9 that can totally make a trade-off between complexity and performance.The structure is similar to the Jinook structure but the first-stage EFM cell replaces with HK-EFM, which can effectively increase the sequence length and lower the whole complexity of hardware implementation, so we denote our structure as HJ-MASH.

Conclusion
A new MASH DDSM (denoted as HJ-MASH DDSM) has been presented to remove spurs in the output spectrum.In addition to the spur-free property, when the number of input bits n is large enough, the deviation between input and output resulted from the feedback loop of first-stage EFM can be ignored, and it can be assumed that the input is approximately equal to the output.Meanwhile, the output sequence length of the proposed structure is  −1 ( − ), the length is higher than Jinook structure and HK-MASH in the worst input case.Besides, the simulation shows that the noise shaping effect of the proposed structure is better than Jinook structure, while almost catch up with HK-MASH, the complexity is decreased and the stability is improved.At the same time, it can be proved that the hardware consumption of the proposed structure is close to the conventional MASH structure, less than HK-MASH [8].
Spur suppression technique are significant to fractional-N frequency synthesizers.Conventional HK-MASH requires extra hardware and has a high complexity to accomplish, and Jinook MASH has a terrible performance while the input is half-scale.The proposed HJ-MASH structure in this paper can perform stable and excellent in any input while just take little hardware overhead as compared with HK-MASH.

Fig. 3 .
Fig. 3. Block diagram of an L th order HK-MASH DDSM incorporating a cascade of HK-EFM1

1 𝑀 2 ( 1 − 𝑧 − 1 ) 2 }𝑋 1 𝑀 2 𝐸 1 (
Fig.8shows the simulated power spectrum of the Jinook MASH DDSM3 when the input is half-scale in the left column and the input is only one-tenth of M in the right column, once again with zero initial condition and no dither.Note that the spectrums of the right column are superior to the left ones, because the Jinook MASH DDSM gets the minimum sequence length and the performance of structure is not stable when the input is equal to half-scale in the left column.Due to the unstable performance, it is necessary to advance a new structure which can ameliorate the stability and do not improve the complexity of structure.

Fig. 9 .
Fig. 9. Block diagram of an L th order proposed HJ-MASH DDSM us derive the sequence length of the proposed HJ-MASH DDSM3.Firstly, the sequence length N 1 of the first stage is exactly equal to ( − ) for all constant inputs and for all initial condition.The sequence lengths of the second and the third stages of the HJ-MASH DDSM3 is  1 •  and  1 •  2 , respectively.Therefore, the sequence length of the L th proposed HJ-MASH DDSM is  1 •  −1 if the signal is constant.In that case, the sequence length of the HJ-MASH DDSM3 is exactly equal to  2 ( − ), which is higher than the sequence length of HK-MASH DDSM3 and Jinook MASH DDSM3 for most input.Therefore, the sequence length of the L th proposed HJ-MASH DDSM is ( − ) •  −1 if the signal is constant.

Fig. 10
Fig. 10 shows the simulated power spectrum of the proposed HJ-MASH DDSM3 when the input is half-scale, once again with zero initial condition and no dither.Note that the spectrum is close to the ideal white noise case, and no individual tones can be seen except the first sub-graph.According to Fig 3, If  is sufficiently small, this pole is very close to the origin in the  plane; equivalently, it is a distant pole which does not significantly affect the overall operation of the modulator.