15-25 MHz Fractional-N Synthesizer

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MASH (Multi-stAge noise SHaping) structure

The MASH and its close relative the sigma delta modulator achieve high resolution in the analogue domain using digital techniques. Commonly found in A/D and D/A converters, as well as frequency synthesizers, they are very popular because it's much easier to make fast, complex digital systems than precision analogue circuitry.

3rd-order is more the norm; but in this project I used a 4th-order (i.e. 4-stage) 32-bit MASH. The yellow delays are latches clocked by the VCO divider:

 Figure 1

The long-term average of output ΔN is exactly F/M, where M is the modulus of the 32-bit overflowing accumulators. ΔN is computed (via a chain of summers, delays and negations) from the carry overflows. When input F is a constant K, the first accumulator overflows K times in every 232 clock cycles, and the long-term mean of ΔN is K/232. Carries from the other accumulators have no long-term effect.

Although ΔN can only take 16 integer values (between -7 and +8 for a 4-stage MASH) the achievable resolution, after low-pass filtering, is extremely high. ΔN is a pseudo-random sequence with peak energy at the Nyquist frequency (fs/2) clearly evident in the time as well as the frequency domain representation:

  4-stage, 24-bit MASH
Figure 2 - ΔN sequence - time & frequency domains [ C++ Code ]
Right: Fourier series of 4-stage, 24-bit MASH; F=1; Hanning Window length = 2^20; Slope = 80dB/decade.

The MASH "colours" its output spectrum. Output noise is concentrated at higher frequencies, where it is easily removed by low-pass filtering. This is termed noise shaping. Each stage cancels its predecessor's noise, substituting new noise of its own. Noise from the final stage remains, but is subjected to several orders of high-pass filtering. This will be demonstrated mathematically using the z-transform. If you've never heard of it, you may need to study elsewhere first; but only a very basic awareness is necessary. Only two z-transforms are actually required:

Transform Description Example
z-1 Unit time delay (one sample period) X[n].z-1 = X[n-1]
1 - z-1 Differentiation (high-pass filter) X[n].(1 - z-1) = X[n] - X[n-1]

Still with me? OK. First, before we get down to the nitty-gritty, a small digression ...

Pascal's triangle

In figure 1, notice how the carry overflows feed a chain of differentiators like this:

  Y[n] = X[n].(1-z-1) = X[n] - X[n-1]

The structure depicted in figure 1 is not the only way to construct a MASH. The stored program which controls this synthesizer is based on an alternative form, which calculates ΔN by a weighted-sum of delayed carries, with weights resembling the rows of Pascal's triangle:

ΔN[n] = C1[n] + C2[n].(1-z-1) + C3[n].(1-z-1)2 + C4[n].(1-z-1)3                 ..... [1]

ΔN[n] = C1[n] + C2[n].(1-z-1) + C3[n].(1-2z-1+z-2) + C4[n].(1-3z-1+3z-2-z-3)     ..... [2]

In equation 1, the chain of differentiators is represented by powers of (1-z-1) which are expanded in equation 2 to reveal the familiar Pascal's triangle pattern:

         +1    -1
      +1    -2    +1
   +1    -3    +3    -1
Terms of the form Ci[n].z-a represent the delayed carries; the negative exponent is the number of clock cycles delayed. Since each row of the triangle, except the first, sums to zero, it's easy to see why only carries from the first stage have a long-term net-effect.

Returning to the main topic ...

Accumulator model

An accumulator is simply a digital integrator:

The challenge comes in modelling carry and overflow. Whenever a carry occurs, we need to subtract 1.0 from the sum:

C[n] ††
0.45 0.45 0.00 -0.45
0.45 0.90 0.00 -0.90
0.45 1.35 1.00 -0.35
0.45 0.80 0.00 -0.80
0.45 1.25 1.00 -0.25
† this can exceed 1.00; it's not the same as the residual sum

†† see below

We can't use non-linear functions like int() and mod() to generate the carry. The equations need to be algebraically manipulable. The standard trick is to inject "quantisation noise" like so:

Figure 3(a) overflowing accumulator; 3(b) 1st order ΣΔ modulator (error feedback topology)

The awkward non-linearity has been absorbed in e[n], which happens to be the negation of the residual sum at the accumulator output. This is very handy, as you will see. You should convince yourself the linearised model on the right is equivalent to the circuit on the left. Also, note the position of the binary point.

We can now derive the transfer function of the stage:

    C[n] = S[n] + e[n]
    S[n] = X[n] - e[n].z-1

Eliminating S[n] yields:

    C[n] = X[n] + e[n].(1-z-1)

This is the transfer function of a first-order sigma delta modulator. Known as the error feedback topology, figure 3(b) is not the only way to draw it. Being a negative feedback system, containing an integrator, it exhibits a high-pass response to the injected noise. The output equals the input plus (shaped) quantisation noise, and the (quantised) output is only 1-bit wide.

MASH Magic

Here comes the best bit. Re-draw the accumulators as a cascade of first-order ΣΔ modulators:

Using the transfer function derived above, we can now write down equations for the overflow carries:
   C1[n] = e1[n].(1-z-1) + F[n]
   C2[n] = e2[n].(1-z-1) - e1[n]
   C3[n] = e3[n].(1-z-1) - e2[n]
   C4[n] = e4[n].(1-z-1) - e3[n]

Finally, we can express output ΔN in terms of fractional input command F:

   ΔN[n] = C1[n] + C2[n].(1-z-1) + C3[n].(1-z-1)2 + C4[n].(1-z-1)3

   ΔN[n] = e4[n].(1-z-1)4 + F[n]

Each stage cancels the noise of its predecessor.

ΔN equals F plus the (highly shaped) noise of the last stage. The high-pass term is raised to the 4th power, and simulated output noise rises at 80dB/decade for a 4-stage MASH, suggesting that e4 is white noise.

Peak-to-peak MASH output increases with increasing MASH order:

en en(1-z-1) en(1-z-1)2 en(1-z-1)3 en(1-z-1)4
0.5 ± 0.5 ±1 ±2 ±4 ±8

MASH output values in the middle of the range (e.g. -1, 0, +1) occur with greatest frequency. Higher outputs (e.g. +8, -7) occur less and sometimes never, depending on F. The probability distribution chart is centred on F.

When a MASH controls a VCO divider, the total number of VCO cycles counted is the sum of all previous MASH outputs. In converting MASH output to divided-VCO edge position at the phase detector, there is an integration which cancels one of the differentiations shown above. This integration at the VCO divider is what converts the N.F frequency command into the phase slope required at the phase detector.

The MASH itself has an ideal step response; however, since it must always be used in conjunction with a low-pass filter to remove the shaped noise, the system step response is that of the filter. Nevertheless, VCO phase can be precisely shifted (an arbitrarily small or large amount) by changing the N.F frequency command for just a single cycle. Time taken to complete the phase adjustment depends on loop bandwidth.