Multirate Digital Signal Processing, Oversampling of Analog-to-Digital Conversion, and Undersampling of Bandpass Signals - Digital Signal Processing: Fundamentals and Applications - page 595

We modify Equation (12.30) into the following useful formats for applications:

f s
2 f max

n ¼ m þ 1 : 5 log 2 0 : 86 (12.31)

f s
2 f max

3

2
3 2 2 ðnmÞ

¼ p (12.32)

EXAMPLE 12.9
Given the following DSP system specifications, determine the equivalent ADC resolution.
Oversampling rate system
First-order SDM with 2-bit ADC
Sampling rate ¼ 4 MHz
Maximum audio input frequency ¼ 4 kHz
Solution:
Since m ¼ 2 bits, and

2f max ¼ 4;000 kHz

f s

2 4 kHz ¼ 500

we calculate

f s
2f max

n ¼ m þ 1:5 log 2 0:86 ¼ 2 þ 1:5 log 2 ð500Þ0:86 z 15 bits

We can also extend the first-order SDM DSP model to the second-order SDM DSP model by
cascading one section of the first-order discrete-time analog filter as depicted in Figure 12.32 .
Similarly to the first-order SDM DSP model, applying the z-transform leads to the following
relationship:

Y z ¼

1 z 1
| {z }
Highpass
noise shaping
filter

2

$ EðzÞ

| {z }
Quantization
error
transform

XðzÞ
| {z }
Original
digital signal
transform

þ (12.33)

Notice that the noise shape filter becomes a second-order highpass filter; hence, the more quantization
noise is pushed to the high frequency range, the better ADC resolution is expected to be. In a similar
analysis to the first-order SDM, we get the following useful formulas:

f s
2 f max

n ¼ m þ 2 : 5 log 2 2 : 14 (12.34)