Multirate Digital Signal Processing, Oversampling of Analog-to-Digital Conversion, and Undersampling of Bandpass Signals - Digital Signal Processing: Fundamentals and Applications - page 593

Discrete-time
analog
integrator

c ()

d ()

d n ( 1 ) area under the curve at n 1

c ()

c () 2

c (1 d n () area under the curve at n

cn ( 1 c ()

c (0

area increment 1() c n

n

012 n 1 n

FIGURE 12.29

Illustration of discrete-time analog integrator.

e () Quantization error

y ()

encoded by m bits

x ()

w ()

1 Decimator

Hz ()

1

1 z

Anti-aliasing LPF

M

1

z

FIGURE 12.30

DSP model for first-order SDM ADC.

After simple algebra, we have
Y z ¼

þ 1 z 1
| {z }
Highpass
filter

XðzÞ
| {z }
Original
digital signal
transform

$ EðzÞ

| {z }
Quantization
error
transform

(12.26)

In Equation (12.26) , the indicated highpass filter pushes quantization noise to the high-frequency
range, where later the quantization noise can be removed by the decimation filter. Thus we call this
highpass filter ð 1 z 1

Þ the noise shaping filter (illustrated in Figure 12.31 ) .
Shaped-in-band noise power after use of decimation filter can be estimated by the solid area under
the curve. We have

U max

2
q
2 p

s 1 e jU

2

Shaped-in-band noise power ¼

dU

(12.27)

U max