1. 蔡宗珉 : Multi-stage Filter Implementation
Class Report
蔡宗珉 :
Multi-stage Filter Implementation

2. Filter Design
Pass Band
Transition
Band
1+ p
1- p
Stop
Band s  p s p
p = ripple (deviation) in the passband from the ideal response
s = ripple (deriation) in the stopband from the ideal response
p = passband edge frequency
s = stopband edge frequency
= p = s
 = p - s
F = fp - fs

3. High Decimation Conversion
LPF
100
10,000 Hz
100 Hz N 5  45 50
5000
F = 50 – 45 = 5 Hz
F = 10,000 Hz
 = 2F/F =
 = 0.001
N  7250
Total multiplications per second:

4. Multi-stage Decimation
LPF1
LPF2
50 2
10,000 Hz
200 Hz
100 Hz N2 N1  45
150
5000 45 50
100
F = 150 – 45 = 105
F = 50 – 45 = 5
N2  150
N1  345
R1  34,500
R2  7,500
Total multiplications per second:
R = R1 + R2 = 42,000

5. Design of practical sampling rate converters
Specify the filter requirements and those for individual stages
Determine the optimum number of stages of decimation or interpolation
Determine the decimation or interpolation factors for each stage
Design an appropriate filter for each stage

6. Multi-stage Interpolation
y[m]
x[n]
h[m] L F FI
x[n]
y[m]
L1
h1(m1)
L2
h2[m2]
LI
hI[mI]

7. Multi-stage Decimation
y[m]
x[n]
Number of multiplications: M
h[m] F0 FI
x[n]
y[m]
h1(m1)
M1
h2(m2)
M2
hI(mI)
MI
FI-1 F0 FI F1
Number of multiplications:

8. Filter Requirements for Individual Stages
y[m]
x[n]
h1(m1)
M1
h2(m2)
M2
hI(mI)
MI
Overall filter requirement for decimation
Passband  0  f  fp
Stage k stopband
 Fk - fs  f  Fk-1/2
k = 1, 2, … , I - 1
Last stage stopband
fs  f  FI-1/2
Passband ripple  p/I
Stopband ripple  s

9. GSM Digital Down Converter
CIC
filter
Half band
filter
Linear phase
filter I
x(n)
cos
C(z) CIC
Decimation
48:1
D(z)
Decimation
2:1
E(z)
Decimation
2:1
DDS Q
sin
KHz
52 MHz
1.08 MHz
KHz
Frequency Characteristics:
H(ej) = C(ej) D(ej) E(ej)
C(ej)
D(ej)
E(ej)
H(ej)
-50
-20
-50
-50
-100
-40
-100
-100
-150
-60
0.5
0.5
0.5
500 KHz

10. CIC Filter CIC Filter: Cascaded Integrator Comb Filter
Purpose: Large sample rate changes of digital systems.
Advantages:
No multiplier required
No memory for coefficients required
Low power consumption
Very regular structure with two simple basic block
Little external control required

11. CIC Building Block
Basic Integrator: A single pole IIR filter with a unity feedback
y[n] = y[n - 1] + x[n]
Basic Comb:
y[n] = x[n] – x[n – M]
Filter Structure:
Z-1
Z-M + - + +
Integrator Element
Comb Element

12. Half Band Filter  The attractive properties is about 50% of the
coefficients are zero.
 The half band filter can be designed with linear phase
under a constraint on its length. Thus, the required
number of multiplications can be reduced to about
25%.
 The pass band and stop band ripples are equal, i.e.
p = s.
 The past band and stop band edges are symmetric
with respect to /2, i.e. p + s = .