Page 290 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL APPLICATIONS PART 2: FRACTIONAL-N FREQUENCY
SYNTHESIZERS Ronald E. Best 172
The signal transfer function is just a delay by one sample. When the input signal is a DC
level, the same DC level will show up at the quantizer output delayed by one sampling interval
(T = 1/f ). This means practically that the input signal is transferred to the output with gain 1.
F
The noise transfer function is simply the transfer function of a digital differentiator (difference
builder)—that is, the output signal at discrete time t = nT is simply the difference e(nT ) − e[(n
− 1)T ]. When calculating the transfer functions as functions of frequency f, we must simply
substitute z with e j2πfT and produce some manipulations

(7.9a)

(7.9b)

We are now able to compute to power spectral density of the quantization error at the output
of the quantizer and digital filter. From this, we can further compute the bit gain of this type of
A/D converter. In the ideal case, the error sequence e(nT) should be a random noise signal (as
shown, for example, in Fig. 7.6c). As we will learn very soon, this is unfortunately not true for
this type of ΣΔ ADC. Later in this section, however, we will introduce some modifications
that will “randomize” the error function so it becomes more or less “white.” Therefore,
let’s assume for the moment that e(nT) is a random sequence; hence, its power density
spectrum becomes flat in the frequency range −f /2 to f 2. The power spectrum of the
F F
quantization error (S ) is depicted in the uppermost trace in Fig. 7.8. To get the power density
ee
spectrum of the quantization error at the output of the lowpass filter, we must multiply S by
ee
the square of |NTF(f)| and by the square of the transfer function of the lowpass filter, which is
denoted here by H (f). Hence, we get
lp

(7.10)

The second trace in Fig. 7.8 shows the squared absolute value of NTF(f), and the third trace
illustrates the squared absolute value of H (f).
lp
Performing the multiplications defined in Eq. (7.10) finally yields the attenuated power
density spectrum S at the output of the digital filter, shown in the last trace of Fig. 7.8. This
yy
result must be compared now with the result obtained for the simple ADC considered earlier
(cf. Fig. 7.6e). Here, the total noise (that is, the area under the S spectrum) was attenuated by
ee
the over-sampling ratio OSR. In the case of the ΣΔ ADC, the attenuation is much larger
because the gain of the differentiator [NTF(f)] is near zero at low frequencies. The bit gain G
of the first-order ΣΔ ADC can be shown to be 55

(7.11)

To get a bit gain of 15, for example, we would have to choose OSR = 1523, which
compares favorably with the result obtained for the previously discussed