Selected Topics in Applied Signal Processing - Digital Signal Processing: An Introduction with MATLAB and Applications - page 173

It is not hard to show that the representation in Fig. 3.35 has an equivalent
input/output relationship to that in Fig. 3.35 ; the former is more hardware efficient.
and so it more frequently used in practice for implementing SDMs. An analysis of
the SDM is performed below. It will be seen that the SDM has the remarkable
property of being able to shape the quantization noise away from the spectral band
occupied by the signal. This means that the SNR in the band of interest can be
made very high.
The SDM can be represented in the s-domain as shown in Fig. 3.36 . The
quantization noise can be represented as additive noise with transfer function N(s).
Since R t 1 r ð t Þ dt L

1

s R ð s Þ [Tables-Laplace Transform Pairs and Theorems], the
integrator is represented as 1/s.
From Fig. 3.36 one can write:

Y ð s Þ¼ E ð s Þ
s

þ N ð s Þ¼ X ð s Þ Y ð s Þ
s

þ N ð s Þ

Hence,

Y ð s Þ¼ 1

1 þ s X ð s Þþ s

1 þ s N ð s Þ¼ H 1 ð s Þ X ð s Þþ H 2 ð s Þ N ð s Þ;

where the signal transfer function H 1 ð s Þ¼ 1
1 þ s

is a LP function, while the noise

transfer function H 2 ð s Þ¼ s

1 þ s is a HP function (see Fig. 3.37 ). Since the input
signal is restricted to having low-frequency content, the band of interest is the low
frequency band. From this it is clear that the SDM passes the signal energy but
attenuates the quantization noise in the band of interest. That is, the noise transfer
function is a HP function, and the quantization noise accumulates outside the band
of interest, as shown in Fig. 3.38 . This highly advantageous property of the SDM
is known as noise shaping.

Fig. 3.36 Sigma-delta
modulation system in the
s-domain

N ( s )

E ( s )

1
s

X ( s ) Y ( s )

Quantizer

[ SDM Encoder ]

Fig. 3.37 Signal and noise
transfer functions in SDM

| H 1 ( f )| | H 2 ( f )|

f f