SUPPRESSING SPURIOUS TONES RESULTING FROM THE REQUANTIZATION OF THE ACCUMULATED QUANTIZATION ERROR WHEN PERFORMING FRACTIONAL FREQUENCY DIVISION

A fractional frequency divider configured to suppress spurious tones resulting from requantization of an accumulated quantization error has a multi-modulus divider (MMD) and one or more digital-to-time converters (DTCs). Each DTC includes a gain and/or digital predistortion, a dithered requantizer having a multibit input signa, a multibit output signal, and a multibit digitally-controllable delay (DCD). The quantized time delay between the input and output of the DTC is determined by the input from the MMD controller, the gain/digital predistortion and the requantizer. The MMD controller provides an input to the DTC that is proportional to the accumulated time quantization error introduced by the MMD controller, the gain/digital predistortion scales by a gain factor to produce a signal (a[k]) that is combined with a discrete-valued dither signal in the dithered requantizer to provide the control input of the DCD.

TECHNICAL FIELD

The disclosure relates to fractional frequency division using a multi-modulus divider (MMD) and a digital-to-time converter (DTC). More particularly, this disclosure relates to generating a DTC output that is free of spurious tones related to digitally-controllable delay (DCD) quantization.

BACKGROUND

Fractional frequency dividers (FFD) are a fundamental component in electronic systems designed to divide the frequency of an input signal by a fractional value. The primary function of an FFD is to generate output frequencies that are non-integer multiples of a given reference frequency. This ability is vital for various applications which require flexible and precise frequency synthesis. FFDs are therefore integral to communication systems and wireless technologies in which the generation of carrier frequencies with high precision is a necessity. Another application of FFDs is in the generation of clock signals used to synchronize the operation of an array of components within a digital system.

A particular application of an FDD is within a Phase-Locked Loop (PLL). The primary function of a PLL is to synchronize the phase of an output signal with the phase of a reference signal. The PLL compares the phases of the input and feedback signals, generating an error signal to adjust the output frequency and phase. The inclusion of an FFD in the PLL provides finer control over the output frequency, enabling the synthesis of frequencies that are non-integer multiples of the reference frequency.

For FFDs using a multi-modulus divider (MMD) that introduces a variable time delay and a digitally-controllable delay (DCD) that modifies said time delay, the difference in resolution between the accumulated quantization error of the MMD controller and DCD input control word is typically resolved by requantization. The process of requantization in the DCD controller within the DTC produces DTC-requantization-related spurious tones in the output spectrum of the DTC. Various methods used and approaches taken to manage the negative effects of DTC-requantization are discussed in this section.

An all-digital phase locked loop (ADPLL) with a multi-modulus divider (MMD) in its feedback path can be used to generate an output frequency that is an integer plus a fraction times a reference frequency, i.e. fout=(Nint+X/M)fref, where Nint, X and M are integers. FIG. 1 shows the block diagram of a typical ADPLL, where Nint, X and M are integers.

The instantaneous divide ratio of the MMD is Ndiv[k]=Nint+y[k], where y is an integer-valued signal produced by a multi-modulus divider controller. The MMD controller, which is typically based on a digital delta-sigma modulator (DΔΣM), is designed such that the signal y that it produces has an average value of X/M and comprises shaped quantization noise. In this way, fout=(Nint+X/M)fref, and the shaped quantization noise is filtered by the lowpass transfer function of the loop.

Quantization noise qdivcon introduced by the MMD controller is accumulated as a time difference Δt between the edges of the reference and divided signals ref(t) and div(t). When the fractional part of the divide ratio is a non-zero constant, the time difference Δt waveform at the input of the time-to-digital converter (TDC) has a characteristic sawtooth shape, as shown in FIG. 2. The said time difference waveform is correlated with the accumulated quantization noise qdivconacc.

The time difference Δt is measured periodically by the time-to-digital converter (TDC), filtered by the digital loop filter (DLF), and ultimately causes unwanted modulation of the control signal that sets the frequency of the digitally-controlled oscillator (DCO). This unwanted modulation manifests itself in the spectrum as excess phase noise and spurious tones (spurs). FIG. 3 shows simulated phase noise and spurs in the output spectrum of the ADPLL of FIG. 1 that result from MMD controller quantization noise. One way to reduce the unwanted modulation is to bring the edges of the output of the MMD into phase alignment with those of the reference signal at the input of the TDC. In this way, the time difference signal Δt[k] that is measured periodically by the TDC would be approximately constant and so would the control input of the DCO. This can be accomplished in principle by using a digitally-controllable delay (DCD) to delay the edge of the MMD output (or the edge of the reference) by a prescribed amount ΔtDTC[k] before applying it to the input of the TDC. The required time delay is related to the accumulated quantization error of the divider controller qdivconacc[k].

FIG. 4A shows an ADPLL with the DTC in series with the MMD in the feedback path, whereby it delays div(t) by an amount ΔtDTC[k] to produce the delayed signal divD(t); FIG. 4B shows an ADPLL with the DTC in series with the reference path delaying ref(t) by an amount ΔtDTC[k] to produce the delayed signal refD(t). FIG. 4C shows an ADPLL with the DTC's delay function implemented by two pieces, denoted DTC1 and DTC2, whereby DTC1 in series with the MMD in the feedback path delays div(t) by an amount ΔtDTC1[k] to produce the delayed signal divD1(t) and DTC2 in series with the reference path delays ref(t) by an amount ΔtDTC2[k] to produce the delayed signal refD2(t). The three architectures in FIG. 4A, FIG. 4B and FIG. 4C are equivalent in terms of implementing a time shift between the edges of the MMD output and the reference signal. Without loss of generality, we will henceforth refer by default to the configuration in FIG. 4A.

If the quantization error introduced by the MMD were cancelled exactly using the DTC, then the output spectrum would be spur-free, as shown in FIG. 5. Note that the simulated phase noise (gray) follows the theoretical prediction (black dashed).

In practice, the quantized delay introduced by the DTC is required to cancel exactly the uniformly quantized accumulated quantization error. The DTC is a nonlinear mixed-signal component whose transfer characteristic from its input (control word) to its output (delay) exhibits nonlinearity that results in nonlinear distortion and consequently inexact cancellation of the divider controller quantization noise.

FIG. 6 shows the output spectrum when the DCD is nonlinear and suffers from a gain error. This causes the output phase noise spectrum to exhibit an elevated low frequency noise floor and spurs.

At a minimum, it is necessary to calibrate the gain of the DCD controller because an error in this gain will result in incomplete cancellation of the MMD controller's accumulated quantization error, even if the DCD control word to delay input-output transfer characteristic were linear. Digital predistortion can be used to compensate for the nonlinearity of the DCD.

FIGS. 7A and 7B show architectures with gain calibration and digital predistortion (DPD).

The number of discrete levels of the accumulated MMD controller quantization error signal qdivconacc may be larger than the number of delay steps available in the DCD. In this case, the DCD controller must perform a requantization. FIG. 8 shows the output spectrum of the ADPLL of FIG. 4A where ideal gain correction and DPD have been applied. Note that the spectrum contains spurious tones; these result from requantization in the DCD controller.

This problem results from the difference between the number of quantization levels of the MMD controller's accumulated quantization noise qdivconacc and the number of quantization levels of the DCD. In particular, the NbDTC-bit delay introduced by the DTC is required to cancel the Nbace-bit accumulated quantization error, where NbDTC<Nbacc. This cannot be realized without introducing a quantization error.

One approach to minimizing nonlinearity-induced spurs resulting from requantization is to use a noise-shaping requantizer, such as a ΔΣ modulator (ΔΣM) instead of a simple quantizer.

FIG. 9 shows a DCD controller containing a first-order ΔΣ modulator to perform the requantization operation.

The ΔΣM scrambles the DCD control word so that it appears noise-like. It does this by pushing the requantization noise to higher frequencies. The reshaped requantization noise appear as excess noise in the output spectrum, as shown in FIG. 10.

The DTC can be implemented with two or more DCDs, as shown in FIG. 11, but this does not solve the requantization problem.

It will be appreciated from FIGS. 8 and 10 that fractional spurs and noise due to requantization in the DTC controller degrade the performance of the overall system in which the fractional divider is being used. This has been found to have a detrimental effect when the system is being used in applications such as communications, radar, and instrumentation. Approaches such as incorporating noise-shaping requantizers replace the unwanted spurs with unwanted noise in the output spectra and so do not solve the problem.

While these examples represent cases where the MMD divider controller receives a constant input, a DTC can be used in combination with an MMD as a fractional divider in more generic cases where the input is time-varying, such as in the case of clocking, communications, radar and signal processing blocks. In these applications too, the presence of nonlinearity may degrade the noise performance with the introduction of spurs and folded noise.

Therefore, in view of the above, there is a need for a method to mitigate DTC-requantization-induced spurs without introducing excessive noise.

SUMMARY

In one aspect of the present invention, there are various embodiments of a fractional frequency divider consisting of a multi-modulus divider (MMD) and a digital-to-time converter (DTC) comprising a dithered requantizer for mitigating spurs induced by the quantization nonlinearity in a DCD controller are disclosed.

In one embodiment, a dithered requantizer for use in a DCD controller is disclosed, wherein:

In one embodiment, the signal s[k] is equal to a[k] and the combination of the dither signal d[k] with s[k] is a simple summation executed by a summer.

In one embodiment, the quantizer which comprises the dithered requantizer is a truncation quantizer, which operates on the summation of a[k] and d[k].

In one embodiment, the quantizer which comprises the dithered requantizer is a rounding quantizer.

In one embodiment, the dither signal d[k] is derived from a random or pseudorandom signal within the system.

In one embodiment, the dither signal d[k] is produced by a random number generator.

In one embodiment, the dither signal d[k] is derived from the quantization error of the MMD controller.

In one embodiment, shown in FIG. 13, the dither signal d[k] is a full-scale uniformly distributed NbΔ-bit random signal produced by a random number generator, where NbΔ=Nbacc−NbDTC.

In one embodiment, the dither signal d[k] is an NbΔ+1-bit random number with a triangular probability density function and a range of 2NbΔ+1.

In one embodiment, the dither signal d[k] is a random number with a range larger than 2NbΔ+1.

In an embodiment, the Nbacc-bit input signal a[k] is partitioned into two signals s[k] and aLO[k] where a[k]=2NbΔs[k]+aLO[k], as shown in FIG. 15, wherein:

In certain embodiments the DCD is partitioned into multiple sections, as shown in FIG. 11 for the case of two sub sections, with DCD1 corresponding to the NbDCD1 most significant bits and DCD2 to the NbDCD2 next most significant bits of qdivconacc In an embodiment, the NbDTC-bit digitally-controllable delay is partitioned into two digitally-controllable delays DCD1 and DCD2 controlled by NbDCD1-bit and NbDCD2-bit signals c1[k] and c2[k] such that NbDTC1+NbDTC2=NbDTC, further comprises:

In an embodiment there is provided fractional frequency divider comprising a multi-modulus divider (MMD) and a digital-to-time converter (DTC) in a series connection, wherein:

In an embodiment there is provided a fractional frequency divider comprising a digital-to-time converter (DTC) wherein the DTC comprises a digital predistortion, a dithered requantizer and a digitally-controllable delay (DCD); the dithered requantizer is configured to combine a discrete-valued dither signal d[k] with a digital signal s[k], and utilize a quantizer to requantize said combination to produce an output digital signal c[k] which is applied to the control input of the DCD.

In an embodiment the digital signal s[k] is generated from a digital signal a[k], where a[k] is a combination of an output signal from a multi-modulus divider (MMD) controller and the digital predistortion associated with the fractional frequency divider.

In an embodiment, the quantization error cancellation function of the DTC can be implemented by inserting the DTC between the MMD output and the div input of the TDC (the “div” path) as shown in FIG. 4A, or between the reference input and the ref input of the TDC (the “ref” path) as shown in FIG. 4B.

In an embodiment, the DTC function is implemented using two DCDs, DCD1 and DCD2 which can be connected in series to form a single effective DTC, as shown in FIG. 11, and inserted in the div or ref paths, as shown in FIG. 4A and FIG. 4B.

In an embodiment, the MMD and DCD1 are connected in cascade in the feedback path and DCD2 is in the reference path, shown in FIG. 4C.

In another aspect of the present invention there is provided a method for suppressing spurious tones in digitally-controllable delay (DCD) systems, the method comprising: generating an accumulated time quantization error signal from a timing signal component, wherein a resolution mismatch between the output of the timing signal component and the quantization levels of the DCD results in a quantization error; scaling the accumulated quantization error signal using a gain and/or digital predistortion to match the DCD input range; combining the scaled signal with a discrete-valued dither signal in a dithered requantizer; requantizing the combined signal in the dithered requantizer to produce a control signal; and applying the control signal to the DCD to generate a quantized time delay.

In an embodiment of the method of for suppressing spurious tones the timing signal component is a multi-modulus divider (MMD) controller.

In another aspect of the present invention, there is provided a method for suppressing spurious tones in a fractional frequency divider comprising a multi-modulus divider (MMD) and a digitally-controllable delay (DCD), the method comprising the steps of:

In another aspect of the present invention there is provided a fractional frequency divider comprising: a multi-modulus divider (MMD),, wherein an MMD controller is configured to generate an accumulated time quantization error signal; a gain and/or digital predistortion module configured to scale the accumulated quantization error signal; a dithered requantizer configured to combine the scaled signal with a discrete-valued dither signal and requantize the combined signal to produce a control signal; and a digitally-controllable delay (DCD) configured to receive the control signal from the dithered requantizer and generate a quantized time delay that suppresses spurious tones.

DETAILED DESCRIPTION

The present disclosure provides a dithered requantizer for a digitally-controllable delay that, among other applications, is suitable for use in series with a multi-modulus divider as an open-loop feedforward fractional divider or as a closed-loop feedback divider in a digital-intensive PLL-based fractional-N frequency synthesizer.

When used in a DCD controller, the dithered requantizer eliminates the spurs that would otherwise arise due to resolution mismatch between the accumulated quantization error introduced by the MMD and the quantization levels of the DCD.

FIG. 1 shows an ADPLL 100 which contains a multi-modulus divider (MMD) 100 in the feedback path, also including a time-to-digital converter (TDC) 104, a digital loop filter (DLF) 103, a digitally controlled oscillator (DCO) 102, which is dithered by an MMD controller 102 to accomplish fractional frequency division.

FIG. 2A illustrates how an output frequency

f
    out
   
   =
   
    
     (
     
      
       N
       int
      
      +
      
       X
       M
      
     
     )
    
    ⁢
    
     f
     ref
    
   
  
  ,

can be realized in the case Nint=3, X=1 and M=4. In this example, the division ratio Ndiv of the MMD 101 is varied by the MMD controller 102 between the values 3 and 4 to realize division by

The top row of FIG. 2A shows the divide ration Ndiv 201, which takes the value Ndiv=3 for three cycles of the reference clock ref(fourth row) in every four and the value Ndiv=4 for one cycle in four, leading to an average value

where E {·} denotes the expectation operator. The output signal out 202 (second row) has a frequency

Due to the changing division ratio Ndiv, the rising edges of the MMD output signal div 203 (third row) are displaced in time relative to the edges of ref. This leads to a varying time difference signal Δt 205, being the difference between the rising edges of the div and ref signals at the input to the time-to-digital converter (TDC).

FIG. 2B shows how the signal Δt 205 is strongly correlated with the accumulated quantization error qdivconacc 206 of the divider (MMD) controller 102.

FIG. 3 shows the presence of spurious tones 301 in the spectrum of the output signal of the PLL in FIG. 1 for an example wherein the MMD controller 102 is a third-order digital ΔΣ modulator. The time difference signal Δt 205 has a characteristic sawtooth shape with apparently random jumps. The accumulated quantization error of the MMD controller 102, qdivconacc 206 matches the time difference signal that causes unwanted modulation of the DCO and spurious tones 301.

This suggests that using qdivconacc ac 206 to control a delay in a DTC can zero the time difference applied at the input of the TDC.

FIG. 4A shows a block diagram of an all-digital phase locked loop 400 with a multi-modulus divider 101 and a digital-to-time converter 401 together implementing fractional division in the feedback path.

FIG. 4B shows a block diagram of an all-digital phase locked loop 410 with a multi-modulus divider 101 in the feedback path and a digital-to-time converter 401 in the reference path together implementing fractional division.

FIG. 4C shows a block diagram of an all-digital phase locked loop 420 with a multi-modulus divider 101 in the feedback path and digital-to-time converters 421, 422 in the feedback and reference paths that together implement fractional division.

The signal qdivconacc 206 can be used to control a digital-to-time converter 401, 421, 422, denoted DTC 401 in FIG. 4A, that delays the div(t) 402 signal from the MMD 101 output by an amount that brings the edges of div 203 and ref 204 into alignment and nominally zeroes Δt 205. Zeroing can be effected by delaying div(t) 402 and/or ref(t) 403, as shown in FIG. 4A, FIG. 4B, and FIG. 4C.

FIG. 5 shows that if Δt 205 could be zeroed exactly, then the envelope of the output phase noise power spectrum of the fractional-N ADPLL could match that of its integer-N counterpart. In practice, qdivconacc 206 is an Nbdivcon-bit signal having 2Nbdivcon levels, while the digitally-controllable delay in the DTC 401 has just 2NbDTC where NbDTC<Nbdivcon. Due to the resolution mismatch between the accumulated quantization error signal qdivconacc 206 and the control signal c applied to the DCD, it is not possible to zero Δt 205.

FIG. 6 shows the excess noise and spurious tones of the output phase noise power spectrum of the fractional-N ADPLL 301 which differs significantly from that of its integer-N counterpart.

In addition to the requantization problem, the DTC 401 is typically nonlinear and a gain factor is required between the accumulated quantization error signal qdivconacc 206 and the control signal c.

FIGS. 7A and 7B show constructions in which gain and nonlinearity correction can be accomplished implementing a programmable gain 705 and/or digital predistortion 707 function between the input to the DCD controller 701 qdivconacc 206 and the input a 706 to the DCD controller's requantizer 704. While these can reduce the excess noise, they cannot remove the spurs 301 that result from requantization in the DCD controller 701.

FIG. 8 shows that the envelope of the output phase noise power spectrum is closer to its integer-N counterpart but it contains spurs 301 resulting from requantization.

FIG. 9 shows one approach to mitigating requantization-induced spurs which is to use a digital ΔΣ modulator 901 to perform the requantization between a 706 and c 703. While this approach succeeds in mitigating the requantization-induced spurs, the digital ΔΣ modulator 901 introduces excess noise.

FIG. 10 shows the spectrum of the output phase noise of the all-digital PLL in FIG. 4A when the gain and nonlinearity errors of the DTC have been corrected using programmable digital gain 705 and/or digital predistortion 706 and the requantizer has been implemented as a digital ΔΣ modulator 901 as in FIG. 9—in this case the spectrum is spur-free but the envelope of the noise 1001 lies above the theoretically achievable performance limit.

FIG. 11 shows a block diagram of a partitioned DTC 1100 comprising two digitally-controllable delays 1101, 1102. The realignment of the ref and div edges can also be accomplished by partitioning the DCD into two digitally-controllable delays, DCD1 1101 and DCD2 1102, that are used to delay one or both of ref(t) and div(t). When the DCD function is partitioned between a coarse delay DCD1 1101 and a fine delay DCD2 1102, the signal qdivconacc 206 is applied as input to the DCD1 controller 1103 controlling the coarse DCD1 1101, and the quantization error of the coarse requantizer is applied as input to the DCD2 controller 1104 controlling the fine DCD2 1102, as shown in FIG. 11. The architecture in FIG. 11 suffers from the same requantization problem as FIG. 4A and FIG. 4B, even when one or more of the requantizers 1105, 1106 is implemented as a digital ΔΣ modulator.

The present disclosure of a dithered requantizer solves the problem of requantization-induced spurs in the DTC and will now be described in conjunction with FIG. 12 onwards.

A conventional DTC 401 comprises a digitally-controllable delay (DCD) 702 and a DCD controller 701. The DCD 702 has a signal input 703, a control input, and a signal output. The NbDTC-bit digital word c 703 applied to the control input sets the quantized time delay between the signal input and the signal output.

The DCD controller 701 takes as input an Nbdivcon-bit digital word qdivconacc 206 corresponding to the accumulated quantization error of a multi-modulus divider controller 102 and outputs the NbDTC-bit digital word c that is applied to the control input 703 of the DCD 702.

The DCD controller 701 performs two functions: it applies gain and/or digital predistortion 705, 706 to the input signal qdivconacc 206 yielding an intermediate signal Nbacc-bit signal a 706, and then requantizes the signal a 706 before outputting the NbDTC-bit digital word c 703.

The dithered requantizer 1201 for use in a DCD controller 701 has an Nbace-bit input signal a[k] 706, an NbDTC-bit output signal c[k] 703, an NbΔ-bit quantizer, where NbΔ=Nbacc−NbDTC, and an additive dither signal d[k] 1301, where the average value of the signal c[k] 703 equals a constant plus the input signal a[k] 706, scaled by ½NbΔ.

FIG. 12 shows an embodiment 1200 in which a dithered requantizer 1201 is implemented within a DCD controller 701 wherein the input-output relation of the dithered requantizer 1201 in the DCD controller 701 is described by the equation:

where Q(·) denotes quantization from Nbacc bits to NbDTC bits, and

where rq is a random number,

E{·} denotes the expectation operator, and where rqos is a constant.

In the Z domain, C(z)=STF(z)A(z)+RQ(z) where A(z), C(z) and D(z) are the Z transforms of a[k] 706, c[k] 703, and d[k] 1301, respectively, STF(z) is a scaling factor and RQ(z) is the Z transform of rq[k].

FIG. 13 shows an implementation in which the dither signal is a full-scale uniformly distributed NbΔ-bit random signal 1301 d[k]=d1[k] produced by a random number generator 1303, where NbΔ=Nbacc−NbDTC, and Q(·) is a truncation quantizer 1302.

FIG. 14A and FIG. 14B show representative graphs of c 703 and E {c} 1401, respectively, versus a 706. While FIG. 14A has a staircase structure due to the resolution mismatch between a 706 and c 703, the averaged DCD control signal E {c} 1401 in FIG. 14B is linearly related to a 706.

FIG. 15 shows an implementation in which the dithered requantizer, Nbacc-bit input signal a[k] 706 is partitioned into two signals s[k] 1501 and aLO[k] 1503 where a[k]=2NbΔs[k]+aLO[k].

aLO[k] 1503 is applied to the input of a Bernoulli binary generator 1502 which produces a one-bit binary output dither signal 1301 d[k]=d2[k]. The Bernoulli binary generator 1502 is designed to output d[k]=1 with a probability P where P=aLO[k]/2NbΔ, and d[k]=0 with a probability equal to 1-P·d[k] 1301 is added to s[k] 1501 to produce the output c[k] 703, where:

Here too,

where rq is a random number,

E and rqos is a constant.

In the case where the DCD 702 is partitioned into coarse and fine sections, as shown in FIG. 11, with DCD1 1101 corresponding to the NbDCD1 most significant bits and DCD2 1102 to the NbDCD2 next most significant bits of qdivconacc 206, a first requantizer, denoted requantizer1 1105 in FIG. 11, is used to quantize the Nbacc-bit signal a1 1107 to produce the NbDCD1-bit control signal c1 1109 that controls DCD1 1101.

The quantization error signal qRQ1 of the first requantizer 1105, being the difference between a[k] 706 and c1[k] 703, is applied to the input of the second DCD controller 1104. The requantizer in the second DCD controller, denoted requantizer2 1106 in FIG. 11, is a dithered requantizer, and is implemented as shown in FIG. 13 or FIG. 15.

The quantization error cancellation function of the DTC 401 can be implemented by inserting the DTC 401 between the MMD output 202 and the div input 203 of the TDC 104 (the “div” path), as shown in FIG. 4A, between the reference input and the ref input of the TDC (the “ref” path), as shown in FIG. 4B, or by inserting two DTCs, one in the div path and the other in the ref path, as shown in FIG. 4C.

When the DTC 401 function is implemented using two DCDs, denoted DCD1 1101 and DCD2 1102, as described above, in one implementation, DCD1 1101 and DCD2 1102 can be connected in series to form a single effective DTC, as shown in FIG. 11, and inserted in the div or ref paths, as shown in FIG. 4A and FIG. 4B. In another implementation, DCD1 1101 could be inserted in the div path and DCD2 1102 in the ref path, as shown in FIG. 4C.

FIG. 16A shows the power spectra of qdivconacc 206 and c 703 when a conventional undithered quantizer with the input-output characteristic of FIG. 14A is employed. Ideally, the spectrum of c 703 should be a scaled version of the spectrum of qdivconacc 206. Due to quantization in the DCD controller 701, the spectrum of c[k] 703 also contains spurs 301. By contrast, FIG. 16B shows that the power spectrum of c[k] 703 is spur-free 1601 in the case of the dithered requantizer 1201 of FIG. 13.

FIG. 17A shows the spectrum of the residual error after cancellation of the MMD's timing error by the DTC 401 with a conventional undithered quantizer 704 in its DCD controller 701. The spectrum contains many spurs 301. By contrast, the spectrum of the residual error after cancellation by the DTC 401 with a dithered quantizer 1201 in its DCD controller 701 is spur-free 1601, as shown in FIG. 17B.