Digital signal processing on arm new

Signal :
How one parameter
Relates to another

voltage
time
continuous signal

continuous
signal
Discrete signal
or
Digitizedsignal
ADC

Y-axis,dependent variable,
range, ordinate, amplitude
X-axis, Independent variable,
domain, abscissa

SampleNumber
100 200 300 400 500 600 700 800 900 1000 1010
N:Total Noof Sample
(1 to N)
(0 to N-1)

Mean (DC)=µ
𝑖=0
𝑁−1
𝑥𝑖
µ =
1
𝑁
𝑥0
+ 𝑥1
+ 𝑥2
+ 𝑥3
… + 𝑥 𝑁 − 1
𝑁
µ =

|𝑥𝑖
− µ|Howfar 𝒊 deviates =
fromthe mean

σ =
𝑖=0
𝑁−1
(𝑥𝑖
− µ) 2
1
𝑁 − 1
𝑥0
− µ 2
+ 𝑥1
− µ 2
+ ⋯ + 𝑥 𝑁 − 1 −
µ 2
(𝑁 − 1)
σ =
P = V 2
Standard Deviation

σ2
= 1
𝑁 − 1 𝑖=0
𝑁−1
(𝑥𝑖
− µ)2
Variance

No DC component :
R .M.S = σ
RMS : Root Mean Square

SNR =
Signal-to-Noise ratio
µ
σ

cv =
Coefficient of Variation
µ
σ X 100

Sample/ Hold :
Quantizer :
Converts dependent
variable from continuous
To discrete
Converts independent
Variable from continuous
To discrete

Ability to reconstruct
exact analog signal
from samples.
Proper Sampling =

Analog
Filter
ADC DSP DAC
Analog
Filter

• Chebyshev
• Bessel
• Butterworth

fc
dB -3dB point
R
C
𝑓
𝑐𝑢𝑡𝑜𝑓𝑓
1
2𝜋𝑅𝐶
=
Frequency (Hz)
RC Passive Low Pass Filter
1 2 3 4 6 7 8
𝑓
= 5 Hz
Passband Stopbandtransition
-3dB = 20 Log(0.707)
to ADC
From sensor
How does this work ?
• At high frequency capacitor
becomes a short circuit
• At low frequency capacitor
is an open circuit

fc
dB
-3dB point
R
C
𝑓
1
2𝜋𝑅𝐶
=
Frequency (Hz)
RC Passive High Pass Filter
From sensor
to ADC
How does this work ?
• At high frequency capacitor
becomes a short circuit
• At low frequency capacitor
is an open circuit

fc
dB -3dB point
Frequency (Hz)
1 2 3 4 6 7 8
Passband Stopbandtransition
fc
dB
Frequency (Hz)
1 2 3 4 6 7 8
Passband Stopband
IDEALPRACTICAL

R
C
Modified Sallen-Key High Pass Filter
R
R f
R 1
C
+
-
𝑘1
𝐶𝑓 𝑐
=R
R1k2=R f

Digital signal processing on arm new

R
C
R
R f
R 1
C
+
-
R
C
R
R f
R 1
C
+
-
R
C
R
R f
R 1
C
+
-
6 POLE BESSEL FILTER
STAGE 1 STAGE 2 STAGE 3

Passband
Stopband
Roll-off
Frequencies NOT allowed to pass
Frequencies allowed to pass
Drop in amplitude

Frequency Response
Chebyshev
Butterworth Bessel

Filter response to rapid
input value change
Step Response =

Ringing and overshoot
Chebyshev
ButterworthBessel

Informationis encoded
in sine waves of signal
Frequencydomain=
Time domain= Informationis encoded
in shape of waveform

How one parameter
Varies with anotherSignal
Output and input
relation.System

Continuous signal : x(t) y(t)
Discrete signal : x[t] y[t]

Linear System :
Homogeneity
Additivity
Shift invariance

x[n] SYSTEM y[n]
SYSTEMkx[n] ky[n]
Homogeneity

x1
[n] SYSTEM
y1
[n]
Additivity
x2
[n] SYSTEM y2
[n]
x1
[n] + SYSTEM y1
[n] +x2
[n] y2
[n]

x[n] SYSTEM y[n]
SYSTEMx[n + s] y[n + s]
ShiftInvariance

Combining signals
through scaling andaddition
SYNTHESIS
Breaking a signal
Into additive components
DECOMPOSITION

+
+
x0
[n]
x1
[n]
x2
[n]
x[n]
SYNTHESIS
DECOMPOSITION

x[n]
DECOMPOSITION
x0
[n]
x1
[n]
x2
[n]
SYSTEM y0
[n]
SYSTEM
SYSTEM
y1
[n]
y2
[n]
y[n]
SYNTHESIS

x[n]
x0
[n] x1
[n] x2
[n] x27
[n]
IMPULSE DECOMPOSITION

x[n]
x0
[n] x1
[n] x2
[n] x27
[n]
STEP DECOMPOSITION

Convolution
Input`
Impulse`
output
*

SYSTEM
δ[n] h[n]
Impulse ResponseDelta Function

1 2 3 4 5 6 7 8 9 10 11
0
1
2
3
4
-4
-3
-2
-1
a[n] = -3δ[n-8]
SYSTEMδ[n] h[n]
SYSTEM-3δ[n-8] -3h[n-8]

x[n] h[n] y[n]=*Input Signal Impulse response Output signal
h[n]]x[n] y[n]
* =

* =
x[n] h[n] y[n]
x[0] h[n-0] x[1] h[n-1] x[2] h[n-2] x[3] h[n-3]
x[5] h[n-5] x[8] h[n-8]x[4] h[n-4] x[6] h[n-6]
0δ[n-0] 0h[n-0]
1.4δ[n-4] 1.4h[n-4] -0.5 δ[n-8] -0.5h[n-8]

* =
x[n] h[n] y[n]
x[0] h[n-0] x[1] h[n-2]
x[3] h[n-3] x[4] h[n-4]

Commutative property of convolution

* =
x[n] h[n] y[n]
x[0] h[n-0] x[1] h[n-1] x[3] h[n-3]x[2] h[n-2]
x[5] h[n-5] x[8] h[n-8]x[4] h[n-4] x[6] h[n-6]
y[6]
y[n] where n= 6

y[6]
x[5] h[n-5]x[4] h[n-4] x[6] h[n-6]x[3] h[n-3]
x[3] x[4] x[5] x[6]
= x[3]h[n-3] x[4]h[n-4] x[5]h[n-5] x[6]h[n-6]+ + +
x[3]h[6-3] x[4]h[6-4] x[5]h[6-5] x[6]h[6-6]+ + +=
= x[3]h[3] x[4]h[2] x[5]h[1] x[6]h[0]+ + +
y[n] where n= 6

y[ i ] =
𝑗=0
𝑀−1
ℎ j x[ i−j ]
Convolution Sum
y[n] =h[n] * x[n]

1. Identity
* δ[n] =𝒙[n] 𝒙[n]
𝒙[n] * kδ[n] = 𝒌𝒙[n]
𝒙[n] * δ[n+s] = 𝒙[n+s]

1. First Differenceand Running sum
𝒚[n] = 𝒙[n] - 𝒙[n-1]
First difference

2. First Differenceand Running sum
𝒚[n] = 𝒙[n] + 𝒚[n-1]
Running Sum
𝒚[21] = 𝒙[21] + 𝒚[20]

3. MathematicalProperties
Commutative
Associative
𝒂[n] * 𝒃[n] = 𝒂𝒃[n] * 𝒂[n]
(𝒂[n] *𝒃[n] )* 𝒄[n] = 𝒂[n]*( 𝒃[n] *c [n])
Distributive
𝒂[n] *𝒃[n] + 𝒂 𝒏 𝒂[n]*( 𝒃[n] +c [n])*𝒄[n] =

Periodic-Discrete
Periodic - Continuous
Aperiodic-Discrete
Aperiodic -Continuous
-Discrete FourierTransform
- FourierSeries
-Discrete Time Fourier Transform
- FourierTransform

Discrete Fourier
Transform( DFT )

N point input DFT
𝑁
2
+ 1 𝑐𝑜𝑠𝑖𝑛𝑒waveamplitudes
𝑁
2
+ 1 𝑠𝑖𝑛𝑒waveamplitudes
Time domain
Frequency domain
𝑅𝑒 𝑋[ ]
Im 𝑋[ ]
x[ ]

Time domain Frequency domain
-Decomposition
-Analysis
-Forward DFT
-DFT
Time domain
-Synthesis
-InverseDFT
Frequency domain
𝑥[ ] X[ ]

• Letnumberofpoints=N
DFT
𝑅𝑒 𝑋[ ]
𝑋[ ]𝑥 [ ]
Im 𝑋[ ]𝑥 0 … 𝑥[𝑁 − 1]
Im 0 … 𝐼𝑚[𝑁/2]
Re 0 … 𝑅𝑒[𝑁/2]

-8
-4
0
4
8
0
-8
-4
0
4
8
-8
-4
0
4
8
0 16 32 48 64
-2
-1
0
1
2
0 16 32 48 64 80 96 112 127
0 16 32 48 64 80
-8
-4
0
4
8
0 0.1𝑓 0.2 𝑓 0.3 𝑓 0.4 𝑓 0.5 𝑓
-8
-4
0
4
8
0 4
5
𝜋 π
-8
-4
0
4
8
3
5
𝜋
2
5
𝜋
1
5
𝜋
SampleNumber
Amplitude
AmplitudeAmplitude
Amplitude
frequency frequency
frequency frequency
frequency frequency
DFT
𝒙 [ ]
𝑅𝑒 𝑋[ ]
𝑅𝑒 𝑋[ ]
𝑅𝑒 𝑋[ ]
Im 𝑋[ ]
Im 𝑋[ ]
Im 𝑋[ ]
k : 0 to
𝑁
2
𝜔 ∶
0 to 0.5f :
0 to 𝜋
1
5
𝜋
2
5
𝜋
3
5
𝜋
4
5
𝜋
0 0.1𝑓 0.2 𝑓 0.3 𝑓 0.4 𝑓

A set of sine and cosine waves with unity
amplitude.
𝑐 𝑘[𝑖]
s 𝑖𝑛( )2𝜋𝑘𝑖
𝑁
𝑠 𝑘[𝑖]
=
=
cos( )
2𝜋𝑘𝑖
𝑁 𝑅𝑒𝑋[𝑘]
𝐼𝑚𝑋[𝑘]
𝑠 𝑘 =
cosine wave for
amplitudeheld at
sine wave for
amplitude held at
𝐶 𝑘 =

𝑆0[ ]
𝑆2[ ] 𝑆10[ ] 𝑆16[ ]
𝐶0[ ] 𝐶2[ ] 𝐶10[ ] 𝐶16[ ]
N= 32
𝑅𝑒𝑋[0] 𝑅𝑒𝑋[2] 𝑅𝑒𝑋[10] 𝑅𝑒𝑋[16]
𝐼𝑚𝑋[0] 𝐼𝑚𝑋[2] 𝐼𝑚𝑋[10] 𝐼𝑚𝑋[16]

Synthesis:
Deducing Inverse DFT

𝑘=0
𝑁/2
𝑅𝑒 𝑋 𝑘 cos( )
2𝜋𝑘𝑖
𝑁
𝑘=0
𝑁/2
𝐼𝑚 𝑋 𝑘 sin( )
2𝜋𝑘𝑖
𝑁
+=𝑥[𝑖]
IDFT
𝑖 ∶ 0 𝑡𝑜 𝑁 − 1

𝑅𝑒 𝑋 𝑘 , 𝐼𝑚 𝑋 𝑘𝐼𝑚 𝑋 𝑘𝑅𝑒 𝑋 𝑘 ,
Frequency domain
signals
Amplitudes needed
for synthesis

𝑅𝑒 𝑋 𝑘 =
𝑁/2
𝑅𝑒 𝑋 𝑘
𝐼𝑚 𝑋 𝑘 =
𝑁/2
𝐼𝑚 𝑋 𝑘
𝑅𝑒 𝑋 0 =
𝑁
𝑅𝑒 𝑋 𝑘
Except :
𝐼𝑚 𝑋 0 =
𝑁
Except :
𝐼𝑚 𝑋 𝑘

IDFT
𝑁/2
𝑅𝑒 𝑋 𝑘
𝑘=0
𝑁/2
𝑅𝑒 𝑋 𝑘 cos( )
2𝜋𝑘𝑖
𝑁
Timedomain ReX[ ] : Frequencydomain
Re X [ ] : cosinewave amplitude

• Simultaneous equations
• Correlation
• FFT

s3[ ] : basis function being soughtx1[ ] : signal being analyzed x1[ ] x s3 [ ]
x2[ ] : signal being analyzed s3[ ] : basis function being sought x2[ ] x s3 [ ]
Sum of all points = 32
Sum of all points = 0

𝑖=0
𝑁−1
cos( )
2𝜋𝑘𝑖
𝑁= 𝑥[𝑖]Re X [k]
𝑖=0
𝑁−1
sin( )
2𝜋𝑘𝑖
𝑁= 𝑥[𝑖]Im X [k] -
DFT equations

Duality
Time
DomainFrequency Domain
Single point Sinusoid
Single pointSinusoid
Convolution
Convolution
Multiplication
Multiplication

𝑅𝑒 𝑋 𝑘 , 𝐼𝑚 𝑋 𝑘
Rectangular Notation
𝑀𝑎𝑔𝑋 𝑘 , 𝑃ℎ𝑎𝑠𝑒𝑋 𝑘
Polar Notation

𝐴𝑐𝑜𝑠 𝑥 + 𝐵𝑠𝑖𝑛 𝑥 = 𝑀𝑐𝑜𝑠(𝑥 + θ)
𝑀 = 𝐴2 + 𝐵2
Θ =
𝐵
𝐴
arctan( )
Θ
A
B
M

𝑀𝑎𝑔 𝑘 =
𝑃ℎ𝑎𝑠𝑒𝑋 𝑘 =
(𝑅𝑒 𝑋 𝑘 2 + 𝐼𝑚 𝑋 𝑘 2
)
𝐼𝑚𝑋[𝑘]
𝑅𝑒𝑋[𝑘]
arctan( )
Rectangular-to-Polarconversion

𝑅𝑒𝑋 𝑘 = 𝑀𝑎𝑔𝑋 𝑘 cos(𝑃ℎ𝑎𝑠𝑒𝑋 𝑘 )
𝐼𝑚𝑋 𝑘 = 𝑀𝑎𝑔𝑋 𝑘 sin(𝑃ℎ𝑎𝑠𝑒𝑋 𝑘 )
Polar-to-Rectangularconversion

Re X[ ] Mag X[ ]
Im X[ ] Phase X[ ]

DFT
Spectral Analysis
Frequency response
Signal processing

Spectral Analysis
Phase
AmplitudeFrequency

X
=
WindowedSignal
HammingWindowMeasuredSignal

WindowedSignal
DFT
( Polar Notation )
Averaging
100Spectra
SingleSpectrum
AveragedSpectrum

Interference signal
Realsignal
Harmonics of Real Signal

FrequencyResponse
Changesin amplitudeandphase
of output cosinewaves.
𝑦 𝑛ℎ 𝑛𝑥 𝑛
𝑋 𝑛 𝐻 𝑛 𝑌 𝑛
𝑥 𝑛 ℎ 𝑛 y 𝑛=∗
𝑋 𝑛 𝐻 𝑛 𝑌 𝑛=X
DFT
IDFT
DFT
DFT
IDFT
IDFT

Impulse Response Frequency Response
Frequency ResponseImpulse Response padded with zeros

-1 - 2j
B =
C =
Real(Re)
-6 +2j
3 + 8jA =
Imaginary(Im)
3 8
-6 2
-1 2

𝑎 + 𝑏𝑗 + 𝑐 + 𝑑𝑗 = (𝑎 + 𝑐) +𝑗(𝑏 + 𝑑)
𝑎 + 𝑏𝑗 − 𝑐 + 𝑑𝑗 = (𝑎 − 𝑐) +𝑗(𝑏 − 𝑑)
𝑎 + 𝑏𝑗 𝑐 + 𝑑𝑗 = (𝑎𝑐 − 𝑏𝑑) +𝑗(𝑏𝑐 + 𝑎𝑑)
Addition
Subtraction
Multiplication

𝑅𝑒𝐴 = 𝑀𝑐𝑜𝑠(θ)
𝑀 = (𝑅𝑒𝐴)2 + 𝐼𝑚𝐴)2
Θ =
𝐼𝑚𝐴
𝑅𝑒𝐴
arctan[ ]
𝐼𝑚𝐴 = 𝑀𝑠𝑖𝑛(θ)
PolarNotation

𝑎 + 𝑗𝑏 = 𝑀(𝑐𝑜𝑠θ + 𝑗 𝑠𝑖𝑛θ)
𝑅𝑒𝐴 = 𝑀𝑐𝑜𝑠(θ)
𝐼𝑚𝐴 = 𝑀𝑠𝑖𝑛(θ)

Euler’s Relation
𝑒 𝑗𝑥
= 𝑐𝑜𝑠𝑥 + 𝑗𝑠𝑖𝑛𝑥

Complex exponential
𝑒 + 𝑗𝑏 = 𝑀𝑒 𝑗θ
𝑒 𝑗𝑥
= 𝑐𝑜𝑠𝑥 + 𝑗𝑠𝑖𝑛𝑥

𝑀1
𝑒 𝑗θ1 𝑀2
𝑒 𝑗θ2 = 𝑀1
𝑀2
𝑒 𝑗(θ1+θ2)
Multiplication

𝑀1
𝑒 𝑗θ1
Division
𝑀2
𝑒 𝑗θ2
𝑀1
𝑀2
[ ]= 𝑒 𝑗(θ1- θ2)

𝐴𝑐𝑜𝑠 ω𝑥 + 𝐵𝑠𝑖𝑛 ω𝑥 𝑎 + 𝑗𝑏
Representation of Sinusoids
𝐴 𝑎
𝐵 − 𝑏
: Amplitude ofcosinewave
: Negative amplitude of sine wave

Representation of Sinusoids
M 𝑀 : Amplitude ofcosinewave
: Negative amplitude of sine wave
𝑀𝑐𝑜𝑠 ω𝑡 + ϕ 𝑀𝑒 𝑗θ
θ − ϕ

LINEAR
SYSTEM
3 𝑐𝑜𝑠 ω𝑡 + π/4 1.5 𝑐𝑜𝑠 ω𝑡 + π/8
3𝑒 𝑗π 4 0.5𝑒 𝑗3π 8 1.5𝑒 𝑗π 8
=x

𝑖=0
𝑁−1
sin( )
2𝜋𝑘𝑛
𝑁= 𝑥[𝑛]Im X [k] -
2
𝑁
𝑖=0
𝑁−1
cos( )
2𝜋𝑘𝑛
𝑁= 𝑥[𝑛]Re X [k]
2
𝑁

Mathematicalequivalence
𝑒 𝑗𝑥
= cos(𝑥) + 𝑗𝑠𝑖𝑛(𝑥)
cos 𝑥 = 𝑒
𝑗𝑥
+ 𝑒
𝑗𝑥
2
sin 𝑥 = 𝑒
𝑗𝑥
− 𝑒
𝑗𝑥
2

cos ω𝑡 = 𝑒 𝑗 ω 𝑡
+ 𝑒
𝑗ω𝑡1
2
1
2
sin ω𝑡 = 𝑒 𝑗 ω 𝑡
𝑒
𝑗ω𝑡1
2
1
2
Sinusoidal complex number

Complex DFT
𝑖=0
𝑁−1
= 𝑥 𝑛 𝑒 𝑗2𝑘π𝑛 𝑁
X [k]
1
𝑁
UsingPolarnotation

InverseComplex DFT
𝑖=0
𝑁−1
X [k]
1
𝑁
UsingPolarnotation

Complex DFT
𝑖=0
𝑁−1
= 𝑥 𝑛 ( cos
2π𝑘𝑛
𝑁
− 𝑗𝑠𝑖𝑛
2π𝑘𝑛
𝑁
)X [k]
1
𝑁
UsingRectangularnotation

Complex DFT
𝑖=0
𝑁−1
= 𝑥 𝑛 ( cos
2π𝑘𝑛
𝑁
− 𝑗𝑠𝑖𝑛
2π𝑘𝑛
𝑁
)X [k]
1
𝑁
𝑖=0
𝑁−1
cos( )
2𝜋𝑘𝑖
𝑁= 𝑥[𝑖]Re X [k]
𝑖=0
𝑁−1
sin( )
2𝜋𝑘𝑖
𝑁= 𝑥[𝑖]Im X [k] -
Real DFT

𝑖=0
𝑁−1
X [k]
1
𝑁ComplexDFTequation
--------------equation(1)
EachDFTcoefficient : 2N+2(N−1)=4N−2
N-PointDFT:4N2 realmultiplicationsandN(4N−2) realadditions

Decimation-in-TimeFFTAlgorithm Example
equation(2)
N = 8

Termswithevenindices:x(0), x(2),x(4),x(6)
equation(3)
equation(4)

Termswithevenindices:x(1), x(3),x(5),x(7)
equation(5)
equation(6)
equation(7)

: Periodicfunctionof kwithNperiods
: k from 0 to 7
equation(2)

Calculating X(k) and X(k+4)E.g.
equation(9)
equation(10)
equation(11)
Since G(k) and H(k) are periodic with 4 :

8 point DFT : 4N2 ,N =8
4(8)2 =256
Two 4-point DFT : 4(4)2 + 4(4)2
64 + 64
= 128
4N extra : (4x4) + (4x4)
Total = 160
=32

N points : X[0] toX[N-1]
ReX(0) ImX(0)

Npointstimedomainsignal
Ntimedomainsignalseach
madeofasinglepoint
Decomposition
1

CalculateNfrequencyspectracorrespondingto
theseNtimedomainsignals.2

SynthesizeNfrequencyspectraintoasingle
frequencyspectrum3

N = 16 Four stage decomposition

Interlaced decomposition
Even samples
Odd samples
N points : Log2N stages of decomposition
E.g. 16 point signal (24) : 4 stages
512 points signal (27) : 7 stages
4096 points signal (212) : 12 stages

Bit reversal
Normal sample order Bit reversed order

Consider two 4-point time
domain signals :
Signal 1 :abcd
Signal 2 :efgh
TimeDomain Frequency Domain

Digital Filters
Signal Separation Signal Restoration

Impulse
Response
StepResponse
Frequency
Response
Linear Filter
FFT

Input`
Impulse`
output
*
Filter Kernel

Decibel
bel : Power change by factor of10
E.g.4 bels = 10x10x10x10 = 10000
decibel(dB) : one-tenth of a bel
-20dB-10dB0dB 10dB 20dB
0.01 0.1 1 10 100 power

dB 10𝑙𝑜𝑔10=
𝑃2
𝑃1
dB 20𝑙𝑜𝑔10=
𝐴2
𝐴1
dB 4.342945𝑙𝑜𝑔 𝑒=
𝑃2
𝑃1
dB 8.685890𝑙𝑜𝑔 𝑒=
𝐴2
𝐴1
𝑙𝑜𝑔 𝑒 𝑥 = 𝐼𝑛 𝑥
A = Amplitude
P = Power
dBV = 1 volt rms signal
-3dB = amplitude reduction of 0.707
power reduction of 0.5

Frequency Response Frequency Response in dB

Signal InformationRepresentation
Time domain information
Frequency domaininformation

Time Domain Parameters
1.Step Response
Slow Step Response Fast Step Response

2.Phase Linearity
Linear Phase Nonlinear Phase

Frequency Domain Parameters
Stopband : Frequenciesblocked
Passband : Frequencies allowedto
pass
Transition band : Frequencies between
passband and stopband
Fast Roll-off : Narrow transmission band

Low-pass
High- pass
Band-pass
Band-reject

Ripples in passband Flat passband

Poor stopband attenuation Good stopband attenuation

Filter Design Using
Spectral Inversion

Low-pass
Filter
Spectral
Inversion
High-pass
Filter
Band-pass
Filter
Band-reject
Filter
Low-pass
Filter
Spectral
Reversal
High-pass
Filter
Band-pass
Filter
Band-reject
Filter

• 51-pointsfilterkernel
• Frequency response found
By adding13 zeros to
filter kernel
• Then taking 64-point FFT
• Sign of each samplein
Filter kernel ischanged
• Oneis addedtothe
Sampleat center of
symmetry
Original filter kernel Original frequency response
Filter kernel with spectral inversion Inverted frequency response

x[n]
h[n]
δ[n]
+ y[n]
All - pass
Low- pass
x[n] h[n]δ[n] y[n]
Designinga high-passfilterby spectralinversion
1
2
High- pass

Spectral
Inversion
High-pass Low-pass
Band-pass Band-reject
Low-pass High-pass
Band-reject Band-pass

Filter Design Using
Spectral Reversal

• 51-pointsfilterkernel
• Frequency response found
By adding13 zeros to
filter kernel
• Then taking 64-point FFT
• Sign of every other
sample in the filter
kernel is changed.
Original filter kernel Original frequency response
Filter kernel with spectral reversal Reversed frequency response

Low- pass
x[n] h2[n]h1[n] y[n]
x[n] h2[n]h1[n] y[n]
*
High- pass
Band- pass
Designinga band-passfilterby spectralreversal
1
2

x[n]
h1[n]
h2[n]
+ y[n]
Low- pass
x[n] h[n]δ[n] y[n]
Designinga bandrejectfilterby spectralreversal
1
2
High- pass
Band-reject

BY METHOD OF IMPLEMENTATION
BYUSE CONVOLUTION
Finite Impulse Response (FIR)
RECURSION
Infinite Impulse Response (FIR)
Moving Average SinglePole
Window-sinc Chebyshev
CustomFIR CustomIIR

Finite ImpulseResponse (FIR) Filters

𝑗=0
𝑀−1
= 𝑥 𝑖 + 𝑗
1
𝑀
𝑦 𝑖
𝑥 𝑖
𝑦 𝑖
𝑀
: 𝐼𝑛𝑝𝑢𝑡 𝑠𝑖𝑔𝑛𝑎𝑙
: 𝑂𝑢𝑡𝑝𝑢𝑡 𝑠𝑖𝑔𝑛𝑎𝑙
: 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑖𝑛𝑡𝑠

𝑦 50 =
𝑥 50 + 𝑥 51 + 𝑥 52 + 𝑥 53 + 𝑥[54]
5
Point 50 in a 5-point moving average
𝑥 78 + 𝑥 79 + 𝑥 80 + 𝑥 81 + 𝑥[82]
5
𝑦 50 =
Usingsymmetrically chosenpoints

Original signal
11- point moving average
51- point moving average

Amount of noise reduction is equal to the square-
root of the number of point averaged.
E.g. 100-point
moving average
Noise reduction by factor of 10

sin(π𝑓𝑀)
𝑀𝑠𝑖𝑛(π𝑓)
𝐻[𝑓] =
𝑀 = Number ofpoints
𝑓 : Runsbetween 0and0.5
𝑊ℎ𝑒𝑛 𝑓 = 0, 𝐻 𝑓 = 1
Frequencyresponse of the Moving-Average

The Multiple-Pass
Moving Average Filter

Filter Kernel
Frequency Response
Step Response Frequency Response (dB)
FFT
Integration 20Log()

The Recursive
Moving Average Filter

𝑥 47 + 𝑥 48 + 𝑥 49 + 𝑥 50 + 𝑥 51 + 𝑥 52 + 𝑥 53𝑦 50 =
𝑥 48 + 𝑥 49 + 𝑥 50 + 𝑥 51 + 𝑥 52 + 𝑥 53 + 𝑥 54𝑦 51 =
𝑦 51 = 𝑦 50 + 𝑥 54 − 𝑥[47]
Passing x[ ] througha7-pointmoving averagetoobtainy [ ]

𝑦 𝑖 = 𝑦 𝑖 − 1 + 𝑥 𝑖 + 𝑝 − 𝑥[𝑖 − 𝑝]
𝑤ℎ𝑒𝑟𝑒 ∶ 𝑝 =
𝑀 − 1
2
𝑞 = 𝑝 + 1
𝑀 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑖𝑛𝑡𝑠 𝑖𝑛
𝑚𝑜𝑣𝑖𝑛𝑔 − 𝑎𝑣𝑒𝑟𝑎𝑔𝑒
The Recursive Moving-Average Algorithm

• Frequency band separation
• Bad time domain performance

Ideal Filter Kernel Ideal Frequency Response
sin(𝑥)
𝑥
sinc function :

sin(𝑥)
𝑥
sinc function :
sin(2π𝑓 𝑐 𝑖)
𝑖π
ℎ[𝑖]=
Ideal Frequency Response

Truncated Sinc Filters
Points truncated to M+1 points, symetrically
chosen around the lobe
Allpoints outside M+1 points are set to zero, i.e.
Ignored.
Entire sequence is shifted to right so that it runs
from 0 to M
1
2
3
Truncated-sinc filter kernel
Truncated-sinc frequency response

The Blackmanor HammingWindow
Blackman or Hamming window

Truncated-sinc filter kernel Blackman or Hamming window
X =
Windowed-sinc filter kernel
Windowed-sinc frequency responseTruncated-sinc frequency response

Hamming Window
𝒘 𝒊 = 𝟎. 𝟓𝟒 − 𝟎. 𝟒𝟔 𝐜𝐨𝐬
𝟐𝝅𝒊
𝑴
Blackman Window
𝒘 𝒊 = 𝟎. 𝟒𝟐 − 𝟎. 𝟓 𝐜𝐨𝐬
𝟐𝝅𝒊
𝑴
+ 𝟎. 𝟎𝟖 𝒄𝒐𝒔
𝟒𝝅𝒊
𝑴
𝑖 ∶ 0 𝑡𝑜 𝑀
M ∶ 𝐿𝑒𝑛𝑔ℎ𝑡 𝑜𝑓 𝑓𝑖𝑙𝑡𝑒𝑟 𝑘𝑒𝑟𝑛𝑒𝑙

Blackman or Hamming window
Hamming
Blackman
Frequency response
Blackman
Hamming
Frequency response (dB)
Hamming
Blackman
𝒘 𝒊 = 𝟎. 𝟓𝟒 − 𝟎. 𝟒𝟔 𝐜𝐨𝐬
𝟐𝝅𝒊
𝑴
𝒘 𝒊 = 𝟎. 𝟒𝟐 − 𝟎. 𝟓 𝐜𝐨𝐬
𝟐𝝅𝒊
𝑴
+ 𝟎. 𝟎𝟖 𝒄𝒐𝒔
𝟒𝝅𝒊
𝑴
Hamming Window
Blackman Window

Designing the Windowed-Sinc
Filter

Requiredparameters
Cutoff frequency ∶
Filter kernel lenght ∶
𝑓 𝑐
𝑀
𝑏𝑒𝑡𝑤𝑒𝑒𝑛 0 𝑎𝑛𝑑 0.5
𝑀
4
𝐵𝑊
𝑀
𝐵𝑊: 𝑊𝑖𝑑𝑡ℎ 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛 𝑏𝑎𝑛𝑑

BW = 0.2
BW = 0.1
BW = 0.02
Roll-off vs. Kernel length
M = 20
M = 40 M = 200

Roll-off vs. Cutoff frequency
fc = 0.05 fc = 0.25
fc = 0.45

𝒉 𝒊 = 𝑲
𝒔𝒊𝒏(𝟐π𝒇 𝒄(𝒊 − 𝑴/𝟐)
𝒊 − 𝑴/𝟐
[ 𝟎. 𝟒𝟐 − 𝟎. 𝟓 𝒄𝒐𝒔
𝟐𝝅𝒊
𝑴
+ 𝟎. 𝟎𝟖𝒄𝒐𝒔
𝟒π𝒊
𝑴
]
Windowed-Sinc filter kernel

Digital signal processing on arm new

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