ANALYSIS
ANALYSIS
DESIGN
FILTERING
FILTERING
DSP Concept Mapping
Output Signal
FILTERING
Convolution | Difference Equation
Chap.2-3
ANALYSIS
DTFT
MAGNITUDE:�Identify frequency of the noise or the unwanted signal.
ANALYSIS
DTFT
MAGNITUDE:�Check which frequencies have been attenuated.
1
PHASE:�Check phase distortion.
2
y=conv(h,x)
y=filter(b,a,x)
FIR: H=fft(h,M)
IIR: [H,w]=freqz(b,a,M)
zplane(b,a)
plot(w,abs(H))
plot(w,angle(H))
DFT
Chap.11
Poles/Zeros
1
2
Chap.9
Phase
Spectrum
DTFT
Chap.7-8
Magnitude
Check Passband & Stopband frequency
1
Check transition bandwidth & ripple
2
Check phase distortion
1
Chap.10
Discrete-time Input Signal
Continuous Signal
Sampling
Chap.1
1
2
Time-domain Analysis
Causality Test
Stability Test
2 No future input or output.
2 BIBO
ANALYSIS
Z-plane
Z-transform
Chap.4
ROC
STABLE: Unit circle inside the ROC
1
CAUSAL: Right sided ROC
2
Impulse Response�(Filter coeff. b when a=1)
Filter �coeff. b&a
The Filter
IIR DESIGN
Bilinear Transformation
Chap.12
Steps
1
2
Obtain the difference Equation.
3
FIR DESIGN
Windowing Method
Chap.13
Steps
1
2
Solve the impulse response:
3
Chap.5-6
Inverse Z-transform
[b,a]=butter(N,Wc)
h=fir1(N,Wc,window)
fvtool(b,a)
LTI SYSTEM (FILTERING)
FILTERING
Convolution | Difference Equation
Chap.2-3
ANALYSIS
ANALYSIS
DESIGN
FILTERING
DSP Concept Mapping
y=conv(h,x)
y=filter(b,a,x)
FIR: H=fft(h,M), w=2*pi*k/M
IIR: [H,w]=freqz(b,a,M/2)
plot(w,abs(H))
plot(w,angle(H))
Chap.5-6
Inverse Z-transform
[b,a]=butter(N,Wc)
h=fir1(N,Wc,window)
fvtool(b,a)
Sampling
Chap.1
1
2
The Filter
FIR DESIGN
Windowing Method
Chap.13
Steps
1
2
Solve the impulse response:
3
Chap.12
Steps
IIR DESIGN
Bilinear Transformation
1
2
Obtain the difference Equation.
3
SPECTRAL ANALYSIS
DTFT
MAGNITUDE:�Identify frequency of the noise or the unwanted signal.
SPECTRAL ANALYSIS
DTFT
MAGNITUDE:�Check which frequencies have been attenuated.
1
PHASE:�Check phase distortion.
2
Poles/Zeros
1
2
Chap.9
1
Phase
Check phase distortion
Chap.10
DFT
Chap.11
DTFT
Chap.7-8
Magnitude
Check Passband & Stopband frequency
1
Check transition bandwidth & ripple
2
SPECTRAL ANALYSIS
FIR: zplane(h,1)
IIR: zplane(b,a)
plot(n,x)
plot(n/Fs,x)
Z-PLANE ANALYSIS
Z-transform
Chap.4
ROC
STABLE: Unit circle inside the ROC
1
CAUSAL: Right sided ROC
2
plot(n,y)
Causality Test
Stability Test
2 No future input or output.
2 BIBO
ANALYSIS
TIME-DOMAIN ANALYSIS
Discrete-Time Signal
| 2.5 | 3.1 | 4 | 2.7 | 1.4 | ] |
| 0 | 1 | 2 | 3 | 4 | |
| 0 | | | | | |
Signal Values
OPERATION ON INDEX
Reordering the samples; �Folding, Shifting, Selection.
OPERATION ON SIGNAL VALUES
Arithmetic operation is done on each similar sample index.
Math term: vector
SIGNAL LENGTH
No of samples counted from the first to the last non-zero samples.
LTI System
a = [1]
Coefficient a
1y[n] = 2x[n] + 3x[n-1] + 2x[n-2]
y[n] y[n-1] y[n-2] x[n] x[n-1] x[n-2]
y[n] + y[n-1] + y[n-2] = x[n] + x[n-1] + x[n-2]
1 5 3
2 3 2
a = [1 5 3]
Coefficient a
b = [2 3 2]
Coefficient b
DIFFERENCE EQUATION
1
2
1. LTI system is represented by DIFFERENCE EQUATION restricted to operations:
i. Delay
ii. Summation to all samples
iii. Weight multiplication
2. The weights are the filter, called:
i. Coefficient a – output samples weights
ii. Coefficient b – input samples weights
3. LTI system can also be represented by IMPULSE RESPONSE, h[n]:
FIR – equals to coefficient b
IIR – Need z-transform to convert the coefficient a & b to h[n]
IMPULSE RESPONSE
h[n]
FIR
h[n] = b
IIR
h[n] a,b
z-transform
🡪 FIR: Convolution, y[n] = x[n]*h[n]
5
3
3
2
System representation based on it’s process
System representation based on it’s filter
Convolution
x[n] = [3 3 1 3] h[n] = [1 2 1]
x[n] 🡪 [ 3 3 1 3 ]
h[n] 🡪 x [ 1 2 1 ]
3 3 1 3
6 6 2 6
3 3 1 3
+
y[n] 🡪 [ 3 9 10 8 7 3 ]
ns = 0
ns = -1
ns = 0+(-1) = -1
Stability and Causality Check
DIFFERENCE EQUATION
IMPULSE RESPONSE
Z-DOMAIN ROC
Checking BIBO for stability is not always easy since we need to take into account all possible inputs. As an alternative, it is easier to check stability on the Impulse Response.
1y[n] = 2x[n] + 4x[n-1]
h[n] = [2 4]
h[n] = ?
1y[n] - 0.25y[n-2] = 1x[n] + 2x[n-1]
Poles (Denominator Roots):
CAUSALITY: No future input/output
STABILITY: BIBO
CAUSALITY: ROC outward
STABILITY: ROC include unit circle
h[n] = 2.5(0.5)nu[n] � - 1.5(-0.5)nu[n]
Z-transform
Inverse Z-transform
Z-Transform
x[n]
y[n]
h[n]
X(z)
Y(z)
H(z)
Impulse Response
System Function
1y[n] = 0.5x[n] + 0.5x[n-1] + 0.2y[n-1]
Y(z)(1 - 0.2z-1) = X(z)(0.5 + 0.5z-1)
Difference Equation
Time-Domain
Z-Domain
0.2
Re
Im
-1
Table 1 : Z-Transform Pair
Table 2 : Z-Transform Properties
System Realization
+
Addition
Multiplication
Delay
convolution
Multiplication
Fourier Transform
Fourier Transform
x(t)
X(F)
Discrete-time Fourier Transform
Discrete Fourier Transform
x[n]
X(f)
x[n]
X[k]
Whole Plot
Shifted Plot
Half Plot
No of samples in 1 second
Frequency Response of a System
Impulse Response
Difference Equation
System Function
Frequency Response
a=[1]
b=[1 2 1]
System Coefficient
Magnitude Response Variations
Filtering
Time Domain
Frequency Domain
Z-Domain
ROC 🡪 Analyse Stability & Causality.
Poles & Zeros 🡪 Analyse and design the frequency response.�
IIR Butterworth Filter Design
| |
1st Order | |
2nd Order | |
3rd Order | |
| |
1st Order | |
2nd Order | |
3rd Order | |
Normalized analog filter
Digital Filter
Step�1
Step�2
Step�3
IIR Butterworth Lowpass Filter
2nd Order
3rd Order
5th Order
9th Order
IIR Butterwort Filter vs FIR Filter
IIR Butterworth LPF
FIR LPF
Characteristic | IIR Butterworth filter | FIR Filter |
| | |
| Half power 🡪 0.7071 | Half magnitude 🡪 0.5 |
Ripple | Gradually decreasing | Fluctuated |
FIR Filter Design
Step�1
Step�2
Step�3
SOLVE THE IMPULSE RESPONSE EQUATION
window | | |
Rectangular | -20.9 dB; (0.0902) | |
Hanning | -43.9 dB; (0.0064) | |
Hamming | -54.5 dB; (0.0019) | |
Blackman | -75.3 dB; (0.0002) | |
1
2
2
2
3
3
1
Tangent Inverse