\ifthenelse{\isodd{\value{page}}} {\newpage{~}} {}
\section{Pattern noise} \label{PAT}
%%%%%%%%%%%%%%%%%%%%%%
Although first-order \ds modulators are significantly simpler to implement in VLSI than higher-order modulators, they are rarely
used due to two problems:
\begin{enumerate}
\item In traditional SC \ds modulators the oversampling ratio is limited by slew-rate effects and finite op-amp gain. As the
noise-shaping is only first-order, the digital resolution will normally not be sufficient high.
\item The quantization error in a first-order \ds modulator is highly correlated with the input signal which may cause problems
due to pattern noise.
\end{enumerate}
The FDSM concept is, so far, based on first- and second-order noise-shaping. Particularly the D flip-flop FDSM solutions are
suitable for very high sampling speed operation, and can therefore provide a high digital resolution even if the noise shaping is
only first-order. In this way problem no.1 may be overcomed. By using multi-bit quantization, implemented either by parallel
modulators, or the modulo-$2^n$ FDSM, problem no.2 can be significantly reduced as the worst-case signal to pattern noise ratio is
reduced by the number of bits in the quantizer. In some applications it will however be desirable to use a single D flip-flop
FDSM with a bit-stream output, and in this case the effect of pattern noise must be carefully analyzed. In this section we will
start by verifying that the FDSM is equivalent to a traditional \ds modulator with respect to pattern noise. However, since the
internal signal range in the FDSM may be much lower than in a traditional \ds modulator, and the range is defined by
frequency ratios and not by voltages, there can be a significant difference in performance.
\subsection{Background}
%%%%%%%%%%%%%%%%%%%%%%%
The basis for \lig{sq1}\ which predicts the resolution of the delta-sigma modulator is based on the assumption that the
quantization noise is white, and thus uncorrelated with the input signal. This assumption is suitable for most busy input signals.
However, for DC or slowly varying inputs, the white-noise model is far from exact as the quantization error will be heavily
correlated with the input signal. When the input signal is DC, the delta-sigma modulator output will bounce between two levels
keeping its mean equal to the input signal. For certain DC input values the output sequence will be repetitive. If the repetition
frequency lies in the signal band, the modulation will be noisy, if not, it will be quiet. In \fig{dc1}\ (top) the input to a
traditional delta-sigma modulator is swept over different DC values and the resulting in-band noise power is measured
\cite{pattern}. The horizontal line represent the calculated in-band noise level given by the white quantization noise model. As
we see from the figure, there are certain input DC values which produce an output noise power far above the calculated level. For
other input values the measured power is far below the white-noise level. This inherent feature is called pattern-noise. From
\cite{pattern} half the total power is found to be in the end peaks while 1/16 in the central ones.
\begin{figure}[htb]
\begin{center} \epsfig{file=fig/walley1.eps,width=11cm} \end{center}
\caption{\small \em Top) Measured DC scan, traditional delta-sigma
modulator. Middle) Theoretical DC model. Bottom) Simulated FDSM frequency scan. Horizontal lines - white noise model. For all
plots - oversampling ratios = 9.14 \label{dc1}}
\end{figure}
Most traditional delta-sigma modulators need to utilize most of their signal range to reduce the effect of internal noise sources.
It is therefore normally necessary to let the signal range overlap many of the pattern noise peaks in the figure. As we soon will
see, the FDSM is equivalent with respect to pattern noise, but here the internal signal range can be very small compared to the
output levels. As the signal range location is defined by the $f_c/f_{clk}$ ratio, we may for small signal ranges, locate the
signal range in a pattern noise valley and achieve a significantly higher resolution than the white-noise model predicts. A
necessary requirement for doing so, is that we have control over practical effects such as temperature drift and aging which
may affect the $f_c/f_{clk}$ ratio.
\begin{figure}[htb]
\begin{center} \epsfig{file=fig/walley2.eps,width=11cm} \end{center}
\caption{\small \em Theoretic DC model for different oversampling ratios, $f_{clk}=8$MHz. Horizontal lines - white noise model.
\label{dc2}}
\end{figure}
The pattern noise picture is a direct function of the oversampling ratio. In \fig{dc2} a theoretical DC scan is carried out for three
different oversampling ratios. We notice that both the heights of each peak and the valley depth is increased as the oversampling
ratio is increased. In addition, the number of visible peaks increases. However, since the widths of the peaks are reduced, the
power in each peak is inversely proportional to the oversampling ratio cubed.
\subsection{A theoretical model for pattern-noise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In \cite{pattern} an analytical model for the output of a first-order delta-sigma modulator with DC input is
given. The analysis is carried out for a continuos-time modulator, which is shown to be equivalent to a discrete-time model. For a
DC input of amplitude $x$, the output components that lie in half the frequency band can be expressed as
\begin{equation}
y_x(t) = x + 2 \sum_{l=1}^{\infty} \frac{\sin (\pi l x)}{\pi l} \cos (2\pi \, \mbox{frac} [lx] t f_{clk}) \label{pa1}
\end{equation}
Where frac$[x]$ is the fractional roundoff of $x$, that is $x$ minus the closest integer to $x$. In Eq.~\ref{pa1} the first part
is recognized to be the input itself, while the sum represents the quantization noise. As we see, the quantization noise consists
of scaled harmonic components with a frequency dependent on the index $l$. For a harmonic component to lie in the signal band
$0$30dB lower than the white noise level.
In many systems it is not possible to keep a fixed $f_c/f_{clk}$ ratio due to temperature drift and process deviations. In this
case the dynamic range may accidentally be located over one of the peaks in the pattern noise landscape, and a significantly
performance reduction will result. One way to overcome this problem may be to use a feedback arrangement where the mean value of
the decimator output is used to tune either $f_c$ or $f_{clk}$ to achieve a proper $f_c/f_{clk}$ ratio. Another solution is to
use a dither signal to smooth out the pattern noise landscape. In this way the peaks are removed at the cost of shallower valleys as
we will see next.
\begin{figure}[htb]
\begin{center} \epsfig{file=fig/walley_zoom1.eps,width=13cm} \end{center}
\caption{\small \em Frequency scan, theoretic model. Zoom of \fig{dc2} (bottom). Short horizontal line in center/top indicates
FDSM dynamic range. Long horizontal line - white noise model. } \label{walley_zoom1}
\end{figure}