Dsp fundamentals part-ii

Digital Signal Processing
Fundamentals
DSP Ksharing: Rajesh Sharma
www.rajeshsharma.co.in sharma.rajesh@gmail.com
DSP is Mathematics, Algorithms & Techniques
Applied to Digital Signals on DSP

18/11/2008 2
www.rajeshsharma.co.inwww.rajeshsharma.co.inwww.rajeshsharma.co.inwww.rajeshsharma.co.in

18/11/2008 3
Course Contents
ADC/DAC and Sampling Theorem (Go To Topic )
– ADC- Sampling Theorem, Quantization
– DAC- Sample Hold, Reconstruction Filter
– Practical Consideration: guard band, anti alias, anti imaging
DT Signal & System Analysis (Go To Topic )
– DTFT, DFT, FFT, Z-Transform
– DT Systems: Difference Equation
– Schur-Cohn Stability
DCT & MDCT (Go To Topic )
– DCT & MDCT
Digital Filter Structures (Go To Topic )
– FIR & IIR digital filters
– Digital Filter Structures
– Quantization Effects
Right Click Mouse to Open Hyper Link
Go To Topic

18/11/2008 4
DSP System: Components
We will analyze the components of DSP System shown below
DSP
Algorithms
on Signals
Band-limiting Image Removal
Signal
Analysis

18/11/2008 5
Analog DigitalSignal
Conversion
Go 2 Top

18/11/2008 6
Motivation
We cannot take all possible values of t, x(t)
Memory is limited -> countably finite no. of states
Computers:
– Flexibility: Software does the digital signal processing
– Take advantage of the full depth & breadth of processing
tools available for this platform
– Processing performance does not vary with temperature or
time
Reproducibility
– No degradation when copying signal

18/11/2008 7
Objective….
The sampling should not yield any loss of the information.
The problem is how to choose the sampling Frequency Fs.
Analog
To
Digital
DSP
Digital
To
Analog
)(txa
][nx ][ny )(tya
A-to-D Conversion D-to-A Conversion
Sampling

18/11/2008 8
Analog to Digital Conversion
ADC
x(t) x[n]
A-to-D conversion constitute of

18/11/2008 9
Sampling Process : Band-Limited Signals
DSP uses uniform sampling
– Amplitude samples are drawn at regular intervals in time / space

18/11/2008 10
Analog Sinusoidal Signal
10
∞<<∞−+Ω= ttACostxa ..)()( θ
Fπ2=Ω
∞<<∞−+= tFtACostxa ..)2()( θπ
HzorsecondpercyclesinfrequencyisF
secondperradianinfrequencyisΩ
PeriodlFundamentaiswhere)()( papa
TtxTtx =+

18/11/2008 11
Time Domain Sampling: Sampling
11
N
k
F
Ff
fF
N
kn
ACos
n
F
F
ACos
FnTACosnxnTx
T
F
FtACostx
s
s
a
s
a
==
+=
+=
+==
=
+=
asrelatedarefrequencyrelativeandfrequencyanalogthe
)
2
(
)2(
)2()()(
givessecond,persamples1rateatsampledwhich when
)2()(
θ
π
θπ
θπ
θπ

18/11/2008 12
Sampling of a sinusoid

18/11/2008 13
Discrete Time Sinusoid
A Discrete Time Sinusoid is periodic only if its frequency f is rational
number
2
1
2
1samplepercycleinfrequencyis
sampleperradianinfrequencyis
)2(][
2&NumberSample:riableinteger vais
..)()(
<<−
<<−
∞<<∞−+=
=
∞<<∞−+=
ff
nfnACosnx
fn
nnACosnx
πωπω
θπ
πω
θω
13
N
k
fnnxNnx ==+ ifonly:allfor)()(i.e

18/11/2008 14
Frequency Domain Aliasing (FDA)
Discrete time sinusoids whose frequencies are separated by
an integer multiple of are identical i.e. Aliases
14
re uniqueπ, which a|ω|frequencyusoid withingcorrespond
f aan alias oπ arencying frequeusoids havAll those
<
>
sin
||sin ω
π2
}
therrom each ouishable fIndistinguences aresoidal SeqThese Sinu
knACosn
kff
k
kFFF
k
k
k
Sok
,....2,1..).....()(x...2 k
0
0 ±±=+=





+=
+=
+=
θωπωω
All Spectra are Aliases
Except One

18/11/2008 15
Fourier Spectrum of Sampled Signal
∑
∑
∞
=
∞
=
−=
−=
=
0
0
)()()(
)()(
)()()(
n
sss
n
s
s
nTtnTxtx
nTttp
tptxtx
δ
δ
{ } ∑∑
∫
∞
−∞=
∞
∞−
−=−∂=
−=
k ss
s
s
T
k
f
T
nTtTFfP
dfPfXfX
)(
2
)(.)(
)()()( 2
1
δ
π
θθπ
∑
∞
−∞=
−=
k ss
s
T
kfX
T
fX )
1
(
1
)(
F.TF.TF.TF.T
F.TF.TF.TF.T
The spectrum of the sampled signal is a
periodic repetition of the original signal
spectrum

18/11/2008 16
Freq Domain Convolution Graphically
θ
∑ −∂ )( snfθ
)( θ−fX
f
fs
fs

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The sampling theorem has been articulated by Nyquist in 1928 and
mathematically proven by Shannon in 1949
Claude Shannon –
father of information theory
Harry Nyquist

18/11/2008 18
Shannon-Nyquist Sampling Theorem

18/11/2008 19
Sampling Frequency (details)
The way for signal reconstruction is to filter the
signal with the frequency less than half of the
sampling rate.
All aliasing components, except original, are outside this
range, and hence correct signal reconstruction is possible
Sampling Theorem:
An analog signal of bandwidth B can be fully reconstructed
from its samples if Sampling Frequency > 2B

18/11/2008 20
aliases
Spectrum View: Over-sampled (fs>>2f0)
500Hz
Original Frequency
Within range
100Hz-100Hz 900Hz 1100Hz

18/11/2008 21
Aliasing and Folding
However if
may also fall within the range [0,fs/2). So
the signal restored will not be the true original one.
Consider this situations:
This situation is called aliasing due to frequency folding
skff +± 0
02 ffs < 0)2/( ffs <
2
0 0
s
s
f
ff <+−<
00 2 fff s << ss fff << 0)2/(

18/11/2008 22
Aliasing/Folding
Spectrum View: Under-Sampling (f0<fs<2f0)
25Hz
62.5Hz25Hz-25Hz 100Hz

18/11/2008 23
aliasing
Spectrum View: Under-Sampling (fs<f0)
20Hz
20Hz-20Hz 40Hz 60Hz 100Hz

18/11/2008 24
1111 2222
ms5.0
2000
1
=
3
2
3
4
f0
signal
f0'
signal
ms2
500
1
=
0
ttxc π4000cos)( = ttxc π1000cos)( =
Original signal Aliased signal
Aliasing Effect in Time Domain
Alias Examples:
-Fan seems Rotating
In Opposite Direction
-Stroboscopic effect
-Lines in video clip of
TV screen

18/11/2008 25
Aliasing Example

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Aliasing in Images

18/11/2008 27
Digital-to-Analog Conversion
DAC
y(n) y(t)
D-to-A conversion is implemented based on the
principle of interpolation:
∑
∞
−∞=
−=
n
snTtpnyty )(][)(
p(t) is the characteristic pulse shape of the converter.

18/11/2008 28
DAC: Digital-to-Analog Conversion
• The operation is equivalent to pass the discrete
sequence through a system, convolution, with
the impulse response of DAC i.e. p(t)
∑
∞
−∞=
−=
n
snTtpnyty )(][)(
Reconstructed
Signal
Discrete Time Signal &
Interpolation Pulse

18/11/2008 29
Interpolation pulses
Zero Order Hold Linear Interpolation
2nd Order Polynomial SINC Pulse

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Zero Order Hold
Very Bad
Reconstruction

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Linear Interpolation
Still Bad
Reconstruction

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Parabolic Interpolation
Still Not OK
Reconstruction

18/11/2008 33
OverSampling & Zero Order Hold
better
Reconstruction

18/11/2008 34
OverSampling & Linear Interpolation
• Ideal band-limited interpolation: the following
pulse shape will result in perfect reconstruction:
Quite close
Reconstruction
Yet Not Perfect
Reconstruction

18/11/2008 35
Shannon’s Solution
tf
tf
tg
s
s
π
π )sin(
)( =
)(][
0|)(~)(|),(][)()()(~
s
ss
nTxnxwhere
ttxtxnTtgnxdtgnTxtx
=
∀=−−=−= ∑∫∑ ττ
{ }
sT
ffXtxFT
2
1
0)()( ≥∀==
•Provided
•-fs/2 •fs/2 •
•f
•G(f)
•FT

18/11/2008 36
Signal Reconstruction – Graphically
•X
•*
•*
•X

18/11/2008 37
SINC Interpolation: Ideal Low Pass Filter
Perfect
Reconstruction

18/11/2008 38
Practical Considerations
Prefiltering – Anti-aliasing filtering (analog filter)
costly to implement sharp analog filters
→ use over-sampling technique for better reconstruction
and guard band provision





Ω>Ω
<Ω<Ω
=Ω
c
c
a TjH
,0
,1
)(
π

18/11/2008 39
A/D Conversion
Sample&
Hold
A/D
Conversion
aliasinganti −
)(txc
)(tha
)(txa )(txo
1 2
)(txa
)(ts
)(txs
)(tho
)(txo
hold
order-zero
C/D Quantizer Coder
)(txo
][ˆ nxB
][ˆ nxB
][nx ][ˆ nx


 <<
=
otherwise
Tt
th
,0
0,1
)(0
↑
0
t
1
T
∑∑
∞
−∞=
∞
−∞=
−⋅=−⋅=
nn
as nTtnxnTttxtx )()()()()( δδ1
2
3
3
∑ −⋅=∗=
n
oso nTthnxtxthtx )()()()()( 0
)(nTxa↑
t
)(txa )(txo
0 T 2T 3T
)]([)(ˆ nxQnx =
BB
XX
22
2 max
1
max
==∆ +
bits,1+B
B
BB aaaaaaaa −−−
⋅++⋅+⋅+⋅−= 2222 2
2
1
1
0
0210 LL
)complements2'(
∆2
∆
2
∆
2
3∆ )(nx
4

18/11/2008 40
Signal-to-Quantizing Noise Ratio
)(log208.1002.6)*12*2log(.10log10 max
102
max
2
2
2
2
dB
X
B
X
SNR
x
xB
e
x
σ
σ
σ
σ
⋅−+==⋅=












=
∆
=
∆
=
∆
≤<
∆
−−=
∫
∆
∆−
−2/
2/
max
222
22
12
2
12
1
2
)(
2
),()(ˆ)(
X
dee
nenxnxne
B
eσ
4
dBB
X
25.16SNRthen,
)inclusion99.93%gives4(.4/take
signalddistribute-Gaussianofcasein the
maxx
−≈
= σσ
(PCM)codingarithmiclog-
codinglinear-
operationrenttranspa],[ˆ][ˆCoding nxnx B⇔
Noise Power

18/11/2008 41
D/A Conversion
Scaled
by Xm
convert to
Impulse
Zero-order
hold
D/A
Conversoin
Compensated
Reconstruction
Filter
][ny
][ny
)(tyo
)(tyo
)(tyr
)( ΩjHo
)(
~
ΩjH r
2/
0
2
sin2
)(
,0
0,1
)(
)()()(
Tj
o
n
oo
e
t
jH
otherwise
Tt
th
nTthnyty
Ω−
⋅
Ω
Ω
=Ω


 <<
=
−⋅= ∑
∑
∑
∑ ∫
∫∑
+Ω=←Ω⋅=
⋅Ω⋅=
⋅−⋅=
−=Ω
ΩΩ
Ω−
Ω−
∞
∞−
−Ω−
∞
∞−
Ω−
k
a
TjTj
n
nTj
o
n
nTjnTtj
o
n
tj
oo
T
k
jjY
T
eYjHeY
ejHny
edtenTthny
dtenTthnyjY
)
2
(
1
)()()(
)(][
)(][
)(][)(
0
)(
π

18/11/2008 42





<Ω
Ω
Ω
=
Ω
Ω
=Ω




<Ω
=Ω
←⋅Ω=
⋅Ω⋅Ω=
Ω⋅Ω=Ω
Ω
Ω
Ω
elsewhere
T
e
T
T
jH
jH
jH
elsewhere
T
T
jH
eYjH
eYjHjH
jYjHjY
Tj
o
r
r
r
Tj
r
Tj
or
orr
,0
,
)2/sin(
2/
)(
)(
)(
~
,0
,
)(
filterioninterpolatideal)()(
)()()(
~
)()(
~
)(
2/ π
π
holdorder-zero--),(
ideal--),(
dcompensate--),(
~
<Ω
<Ω
<Ω
jH
jH
jH
o
r
r
0
T
T
π
T
π
−
T
π2

18/11/2008 43
DAC: Zero Order Hold
zero order hold DAC filtering, convolution, results spectral
distortion and high frequency images…
•Provision of guard band & postfiltering
•Gives proper reconstruction
Digital to Analog
Converter
Anti Imaging
Post filter
Images
correct
Reconstruction Filter

18/11/2008 44
Multirate Processing for A/D and D/A Conversions
Low-order
Anti-aliasing
Filter
A/D
conversion
Sharp
Lowpass
Filter
a aaab d1c 2c
M↓
][nx][ˆ nx)(txa
)(txc
doversample digital
(1)Oversampling-Decimation based A/D Conversion
(2) Interpolation - Low-order Reconstruction based D/A conversion
L↑
)(tyr][ny ][ˆ ny D/A
Conversion
Simple
Reconstruction
Filter
a aaab c d
)(tyDA
filter
imaginganti −

18/11/2008 45
Over Sampling-Decimation based A/D Conversion
Low-order
Anti-aliasing
Filter
A/D
conversion
Sharp
Lowpass
Filter
a aaab d1c 2c
M↓
][nx][ˆ nx)(txa
)(txc
a
b
d
CN Ω=ΩΩ=Ω
−
stoppass ,
LPFanalogorderLow
signalanalogFilteredLP
1c
2c
sampling)(over
(A/D)Sampling
NCs Ω+Ω=Ω
M
π
ω =N
LPFdigitalorder-High
signal1):(MdecimatedandFiltered
doversample digital

18/11/2008 46
Interpolation - Low-order Reconstruction based D/AC
1b
2b
L↑
)(tyr][ny ][ˆ ny
a
d
CN
T
Ω−=ΩΩ=Ω
−
''
2
,
LPFanalogorderLow
stopp
π
SignalOutputtedReconstruc






=
filterioninterpolat
dcompensate
LPFdigitalorder-High
N
L
π
ω
signalL):(1edInterpolat
D/A
Conversion
Simple
Reconstruction
Filter
a aaab c d
)(tyDA
filter
imaginganti −
c
convertedD/AbetoSignal

18/11/2008 47
Discrete Time
Signal & System Analysis
Go 2 Top

18/11/2008 48
Fourier analysis - tools
Input Time Signal Frequency spectrum
∑
−
=
−
⋅=
1N
0n
N
nkπ2
j
es[n]
N
1
kc~
Discrete
DiscreteDFSDFSPeriodic
(period T)
ContinuousDTFT
Aperiodic
DiscreteDFTDFT
nfπ2je
n
s[n]S(f) −⋅
∞+
−∞=
= ∑
0
0.5
1
1.5
2
2.5
0 2 4 6 8 10 12
time, tk
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 7 8
time, tk
∑
−
=
−
⋅=
1N
0n
N
nkπ2
j
es[n]
N
1
kc~
dt
tfπj2
es(t)S(f)
−∞+
∞−
⋅= ∫
dt
T
0
tωkjes(t)
T
1
kc ∫
−⋅⋅=Periodic
(period T)
Discrete
ContinuousFTFTAperiodic
FSFS
Continuous
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 7 8
time, t
0
0.5
1
1.5
2
2.5
0 2 4 6 8 10 12
time, t
Note: j =√√√√-1, ωωωω = 2ππππ/T, s[n]=s(tn), N = No. of samples

18/11/2008 49
Frequency Domain Sampling: DFS
A Discrete Time Periodic Signal of fundamental period N may
consist of frequency components separated by
The Fourier Series representation of the discrete time periodic
signal will contain at the most N frequency components…
Discrete Time Fourier Series (DTFS): DFS
49
N
f
N
1or2 == πω
1,......1,0for][
1
0
/2
−== ∑
−
=
−
NkenxC
N
n
Nknj
k
π

18/11/2008 50
Discrete Fourier Series (DFS)
N consecutive samples of s[n]N consecutive samples of s[n]
completely describe s in time orcompletely describe s in time or
frequency domains.frequency domains.
DFS generate periodic ck
with same signal period
∑
−
=
⋅=
1N
0k
N
nk2π
j
ekcs[n] ~
Synthesis: finite sum ⇐ band-limited s[n]
Band-limited signal s[n], period = N.
mk,δ
1N
0n
N
-m)n(k2π
j
e
N
1
=
−
=
∑
Kronecker’s delta
Orthogonality in DFS:
synthesis
synthesis
∑
−
=
−
⋅=
1N
0n
N
nk2π
j
es[n]
N
1
kc~
Note:Note: ck+N = ck ⇔⇔⇔⇔ same period N
i.e. time periodicity propagates to frequencies!i.e. time periodicity propagates to frequencies!
DFS defined as:DFS defined as:
~~~~
analysis
analysis

18/11/2008 51
DFS analysis
DFS of periodic discreteDFS of periodic discrete
11--Volt squareVolt square--wavewave























⋅
−
−
±+=
=
otherwise,
N
kπ
sin
N
kLπ
sin
N
N
1)(Lkπ
j
e
2N,...N,0,k,
N
L
kc~
0.6
0 1 2 3 4 5 6 7 8 9 10 k
1
0 2 4 5 6 7 8 9 10 n
θk
-0.4π
0.2
0.24 0.24
0.6 0.6
0.24
1
0.24
-0.2π
0.4π
0.2π
-0.4π
-0.2π
0.4π
0.2π
0.6 ck
~
am
plitude
am
plitude
phase
phase
Discrete signalsDiscrete signals ⇒⇒⇒⇒⇒⇒⇒⇒ periodic frequency spectra.periodic frequency spectra.
Compare to continuous rectangular function
-5 0 1 2 3 4 5 6 7 8 9 10 n
0 L N
s[n]
1
s[n]: period NN, duty factor L/NL/N

18/11/2008 52
DFS properties
Time FrequencyTime Frequency
Homogeneity a·s[n] a·S(k)
Additivity s[n] + u[n] S(k)+U(k)
Linearity a·s[n] + b·u[n] a·S(k)+b·U(k)
Multiplication * s[n] ·u[n]
Convolution * S(k)·U(k)
Time shifting s[n - m]
Frequency shifting S(k - h)
∑
−
=
⋅
1N
0h
h)-S(h)U(k
N
1
∑
−
=
−⋅
1N
0m
m]u[ns[m]
S(k)e T
mk2π
j
⋅
⋅
−
s[n]T
th2π
j
e ⋅
+

18/11/2008 53
Discrete Time FT (DTFT)
Holds for aperiodic signals
∑
−
=
−
⋅=
1N
0n
N
nkπ2
j
es[n]
N
1
kc~
n
s[n] 1 period
n
s[n]
∫⋅=
2π
0
nfπ2j
dfS(f)e
2π
1
s[n]
synthesis
synthesis
nfπ2j
n
es[n]S(f) −
+∞
−∞=
⋅= ∑analysis
analysis
Obtained from DFS as N → ∞
DTFT defined as:DTFT defined as:

18/11/2008 54
DTFT - Convolution
Digital Linear Time Invariant systemDigital Linear Time Invariant system: obeys superposition principle.
∑
∞
=
⋅−=∗=
0m
h[m]m]x[nh[n]x[n]y[n]x[n] h[n]
ConvolutionConvolution
X(f) H(f) Y(f) = X(f) · H(f)
DIGITAL LTI
SYSTEM
h[n]
x[n] y[n]
h[t] = impulse response
DIGITAL
LTI
SYSTEM
0 n
δ[n]
1
0 n
h[n]
0 f
DTFT(δ[n])
1

18/11/2008 55
DTFT - Sampling/convolution
s[n] * u[n] ⇔⇔⇔⇔ S(f) · U(f) ,
s[n] · u[n] ⇔⇔⇔⇔ S(f) * U(f)
(From FT properties)
Time Frequency
t f
s(t) S(f)
t f
ts fsu(t) U(f)
n f
s”[n] S”(f)
Sampling s(t)
Multiply s(t) by
Shah = Щ(t)

18/11/2008 56
Discrete FT (DFT)
∑
−
=
⋅=
1N
0k
N
nk2π
j
ekcs[n] ~synthesis
synthesis
DFT defined as:DFT defined as:
Note:Note: ck+N = ck ⇔⇔⇔⇔ spectrum has period N
~~~~
∑
−
=
−
⋅=
1N
0n
N
nk2π
j
es[n]
N
1
kc~analysis
analysis
Applies to discrete time and frequency signals.
Same form of DFS but for aperiodic signals:
signal treated as periodic for computational purpose only.
DFT bins located @ analysis frequencies fm
DFT ~ bandpass filters centred @ fm
Frequency resolution
Analysis frequencies fAnalysis frequencies fmm
1N...20,m,
N
fm
f S
m −=
⋅
=

18/11/2008 57
Basis Vectors of DFT
•DFT & IDFT
•Can be written as
xFiX
xFX
NxxxxX
ni
n
N
rr
rr
•=
•=
−++++= −−−−
..1,
..1,2
)1(21
)(
)1(
)1(...)2()1()0(.1)1( ξξξ
•DFT projects x[n] on the basis vectors formed by the rows of F

18/11/2008 58
DFT - pulse & sinewave
ck = (1/N) e-jπk(N-1)/N sin(πk)/ sin(πk/N)~~
a) rectangular pulse, width Na) rectangular pulse, width N
r[n] =
1 , if 0≤n≤N-1
0 , otherwise
b) real sinewave, frequency fb) real sinewave, frequency f00 = L/N= L/N
cs[n] = cos(j2πf0n)
ck = (1/N) ejπ{(Nf0-k)-(Nf0 -k)/N} (½) sin{π(Nf0-k)}/ sin{π(Nf0-k)/N)} +
(1/N) ejπ{(Nf0+k)-(Nf0+k)/N} (½) sin{π(Nf0+k)}/ sin{π(Nf0+k)/N)}
~~
i.e. L complete cycles in N sampled points
-5 0 1 2 3 4 5 6 7 8 9 10 n
0 N
s[n]
1
0 1 2 3 4 5 6 7 8 9 10 k
1 1
ck
~

18/11/2008 59
DFT Examples
DFT plots are sampled
version of windowed
DTFT

18/11/2008 60
Linearity a·s[n] + b·u[n] a·S(k)+b·U(k)
Multiplication s[n] ·u[n]
Convolution S(k)·U(k)
Time shifting s[n - m]
Frequency shifting S(k - h)
∑
−
=
⋅
1N
0h
h)-S(h)U(k
N
1
∑
−
=
−⋅
1N
0m
m]u[ns[m]
S(k)e T
mk2π
j
⋅
⋅
−
s[n]T
th2π
j
e ⋅
+
DFT properties
Time FrequencyTime Frequency

18/11/2008 61
DTFT vs. DFT vs. DFS
t
0 T /2 T 2T f
s[n] S(f)
f
~
cK
t
s”[n] D FTIDFT
(a)
(a) Aperiodic discrete signal.
(b)
(b) DTFT transform magnitude.
(c)
(c) Periodic version of (a).
(d)
(d) DFS coefficients = samples of (b).
(e)
(e) Inverse DFT estimates a single period of s[n]
(f)
(f) DFT estimates a single period of (d).

18/11/2008 62
Time Domain Aliasing
If we take N point DFT, i.e. frequency domain sampling, of
a signal X(n) with periodicity L then x(n) can be recovered
only if N >= L otherwise if N<L then there will be time
domain aliasing.
Another reason of Time Domain aliasing is Filtering in
frequency domain or circular convolution in time domain,
without sufficient zero padding in time domain
62
Under Sampling in any Domain will Cause
Aliasing in another domain

18/11/2008 63
Time Domain Aliasing
Linear Convolution
Circular Convolution

18/11/2008 64
TDA vs FDA
64

18/11/2008 65
DFT – leakage
Spectral components belonging to frequencies between twoSpectral components belonging to frequencies between two
successive frequency bins propagate to all bins.successive frequency bins propagate to all bins.
LeakageLeakage
Ex: 32-bins DFT of 1 VP sinusoid sampled @ 32kHz. 1 kHz frequency resolution.
(b)
(b) 8.5 kHz sinusoid
(c)(c) 8.75 kHz sinusoid
(a)
(a) 8 kHz sinusoid
* N·Magnitude
*

18/11/2008 66
1. Cosine wave
DFT - leakage example
s(t) FT{s(t)}
2. Rectangular window4. Sampling function1. Cosine wave
0.25 Hz Cosine wave
3. Windowed cos wave5. Sampled windowed
wave
Leakage caused by sampling for a nonLeakage caused by sampling for a non--integer number of periodsinteger number of periods
s[n] · u[n] ⇔⇔⇔⇔ S(f) * U(f)
(Convolution)

18/11/2008 67
2. Rectangular window4. Sampling function1. Cosine wave
1. Cosine wave3. Windowed cos wave5. Sampled windowed
wave
DFT - coherent sampling
s(t) FT{s(t)}
Coherent sampling: NC input cycles exactly into NS = NC (fS/fIN) sampled points.
s[n] ·u[n] ⇔⇔⇔⇔ S(f) * U(f)
(Convolution)
0.2 Hz Cosine wave

18/11/2008 68
Significance of F/Fs: Harmonic Signals
Coherent sampling of harmonically related signals
– Find F, the fundamental frequency of signal
– Sample at Nyquist rate Fs, considering maximum harmonic
– DFT of integer number of cycles only
Significance of f = F/Fs = k/N
For coherent sampling: Take N point DFT
k signifies the number of integer periods of waves
N signifies the number of samples in those periods of waves
For N point DFT
k signifies the bin number of fundamental
N signifies the minimum point DFT to avoid leakage

18/11/2008 69
DFT – leakage notes
1. Affects Real & Imaginary DFT parts magnitude & phase.
2. Has same effect on harmonics as on fundamental frequency.
3. Affects differently harmonically un-related frequency components of
same signal (ex: vibration studies).
4. Leakage depends on the form of the window (so far only rectangular
window).
LeakageLeakage

18/11/2008 70
DFT – Window characteristics
• Finite discrete sequence ⇒⇒⇒⇒⇒⇒⇒⇒ spectrum convoluted with rectangular window spectrum.
• Leakage amount depends on chosen window & on how signal fits into the window.
Resolution: capability to distinguish different tones. Inversely proportional to main-lobe
width. Wish: as high as possible.Wish: as high as possible.
(1)
(1)
Several windows used (Several windows used (applicationapplication--
dependentdependent): Hamming, Hanning,): Hamming, Hanning,
Blackman, Kaiser ...Blackman, Kaiser ...
Rectangular window
Peak-sidelobe level: maximum response outside the main lobe. Determines if
small signals are hidden by nearby stronger ones.
Wish: as low as possible.Wish: as low as possible.
(2)
(2)
Sidelobe roll-off: sidelobe decay
per decade. Trade-off with (2).
(3)
(3)

18/11/2008 71
Sampled sequence
In time it reduces end-
points discontinuities.
Non
windowed
Windowed
DFT of main windows
Windowing reduces leakage by
minimising sidelobes magnitude.
Some window functions

18/11/2008 72
DFT - Window choice
Window type -3 dB Main-
lobe width
[bins]
-6 dB Main-
lobe width
[bins]
Max sidelobe
level
[dB]
Sidelobe roll-off
[dB/decade]
Rectangular 0.89 1.21 -13.2 20
Hamming 1.3 1.81 - 41.9 20
Hanning 1.44 2 - 31.6 60
Blackman 1.68 2.35 -58 60
Common windows characteristics
NB: Strong DC component can shadow nearby small signals. RemoveNB: Strong DC component can shadow nearby small signals. Remove it!it!
Far & strong interfering components ⇒⇒⇒⇒⇒⇒⇒⇒ high roll-off rate.
Near & strong interfering components ⇒⇒⇒⇒⇒⇒⇒⇒ small max sidelobe level.
Accuracy measure of single tone ⇒⇒⇒⇒⇒⇒⇒⇒ wide main-lobe
Observed signalObserved signal Window wish listWindow wish list

18/11/2008 73
DFT - Window loss remedial
Smooth dataSmooth data--tapering windows cause information loss near edges.tapering windows cause information loss near edges.
• Attenuated inputs get next
window’s full gain & leakage
reduced.
• Usually 50% or 75% overlap
(depends on main lobe width).
Drawback: increased total
processing time.
Solution:
sliding (overlapping) DFTs.
2 x N samples (input signal)
DFT #1
DFT #2
DFT #3
DFT AVERAGING

18/11/2008 74
-1
-0.5
0
0.5
1
0 3 6 9 12 15 18 21 24 27 30
time
0
2
4
6
8
0 3 6 9 12bin
Zero padding
Improves DFT frequency inter-sampling spacing (“resolution”).
-1
-0.5
0
0.5
1
0 4 8 12 16
time
0
2
4
6
8
0 1 2 3 4 5 6bin
0
2
4
6
8
0 12 24 36 48bin
-1
-0.5
0
0.5
1
0 10 20 30 40 50 60 70 80 90 100 110 120
time
After padding bins @ frequencies NS = original samples, L = padded.
LN
fm
f
S
S
m
+
⋅
=

18/11/2008 75
Zero padding -2
• Additional reason for zero-padding: to reach a power-of-two input samples
number (see FFT).
• Zero-padding in frequency domain increases sampling rate in time domain. Note: it
works only if sampling theorem satisfied!
DFT spectral resolutionDFT spectral resolution
Capability to distinguish two closely-
spaced frequencies: not improvednot improved by zero-
padding!.
Frequency inter-sampling spacing:
increasedincreased by zero-padding (DFT
“frequency span” unchanged due to
same sampling frequency)
Apply zero-padding afterafter windowing (if any)! Otherwise
stuffed zeros will partially distort window function.
NOTE

18/11/2008 76
DFT - scalloping loss (SL)
Worst case when f0 falls
exactly midway between 2
successive bins (|r|=½)
SL
bin, k
Note:Note: Non-rectangular windows broaden
DFT main lobe ⇒⇒⇒⇒ SL less severeSL less severe
Correction depends on window used.Correction depends on window used.
f0
Frequency error: εf = r fS/N,
relative error: εR=εf / f0 = r/[(kmax+r)] εR
≤≤≤≤ 1/(1+2 kmax)
kmax
kmax
bin, k
We’re lucky here!
May impact on data
interpretation (wrong f0!)
Input frequency f0 btwn. bin
centres causes magnitude loss
SL = 20 Log10(|cr+kmax /ckmax|)
~~~~
|r| ≤ ½f0 = (kmax + r) fS/N

18/11/2008 77
DFT - SL Example
0 50 100 150 200 250 300 350 400 450 500
0
0.5
1
1.5
2
DC bias correction,
Rectang. window,
zero padding, FFT
DC bias correction,
Hanning window, zero
padding, FFT
• increasing N (?)
• improve windowing,
• zero-padding,
• interpolation around kmax.
SL remedialSL remedial
0 50 100 150 200 250 300 350 400 450 500
0
0.25
0.5
0.75
1

18/11/2008 78
DFT - parabolic interpolation
Parabolic interpolation often enough to find position of
peak (i.e. frequency).
Other algorithms available depending on data.
198 199 200 201 202 203
1.962
1.963
1.964
1.965
1.966
1.967
1.968
199 200 201 202 203 204
0.974
0.975
0.976
0.977
Rectangular window Hanning window

18/11/2008 79
DFT averaging
Incoherent averaging
M
(k)DFT(k)DFT
(k)DFT M1
INCOH
++
=
L M = No. of DFT to averageM = No. of DFT to average
k = bin number, k=0, 1 .. Nk = bin number, k=0, 1 .. N--11
{ } { } { } { }
M
(k)DFTIm(k)DFTIm
j
M
(k)DFTRe(k)DFTRe
(k)DFT M1M1
COH
++
⋅+
++
=
LL
Coherent averaging
Background noise fluctuations reduced, average noise power unchanged.
Background noise power is reduced.

18/11/2008 80
Efficient DFT calculation: FFT
∑∑
−
=
−
=
−
⋅=⋅=
1N
0n
kn
N
1N
0n
N
nk2π
j
k Ws[n]
N
1
es[n]
N
1
c~
WN
n,k = twiddle factors
k = 0,1 .. N-1
DFT
Direct DFT calculation redundancy
WN
kn periodic function calculated many times.
W8
0,k
W8
4,k
W8
2,k
W8
5,k
W8
6,k
W8
3,k
W8
7,k
W8
1,k
Algorithms (= Fast Fourier Transform) developed to compute N-points DFT with ~
Nlog2N multiplications (complexity O(Nlog2N) ).
Direct DFT calculation requires ~N2
complex multiplications.
complexity O(N2)
VERY
BAD
VERY
BAD
!!

18/11/2008 81
FFT
If N is a power of 2, we may
write
),(),(),( 2/
2
2/ oddeven xxx kDFTekDFTkDFT N
N
kj
NN
π−
+=
•Twiddle factor
•N/2 point
• DFT
•x(0)
•x(2)
•x(6)
•x(4)
•Xeven(0)
•Xeven(1)
•Xeven(3)
•Xeven(2)
•N/2 point
• DFT
•x(1)
•x(3)
•x(7)
•x(5)
•Xodd(0)
•Xodd(1)
•Xodd(3)
•Xodd(2)
•w8
0
•w8
1
•w8
2
•w8
3
•w8
4
•w8
5
•w8
6
•w8
7

18/11/2008 82
FFT advantages
0
500
1000
1500
2000
0 10 20 30 40
Number of samples, N
NumberofOperations
DFTDFT ∝∝ NN22
FFTFFT ∝∝ N logN log22NN
512010485761024
44816384128
80102432
4164
Radix-2DFTN
DSPs & PLDs influenced algorithms design. ‘60s & ‘70s: multiplication counts
was “quality factor”. Now: number of additions & memory access (s/w) and
communication costs (h/w) also important.

18/11/2008 83
FFT philosophy
General philosophy (to be applied recursively): divide & conquerdivide & conquer.
Σcost(subsub--problemsproblems) + cost(mappingmapping) < cost(original problemoriginal problem)
Different algorithms balance costs differently.
Example: Decimation-in-time algorithm
time frequency
Step 3: Frequency-domain synthesis.
N spectra synthesised into one.
Step 2: 1-point input spectra calculation. (Nothing to
do!)
Step 1: Time-domain decomposition. N-points signal →
N, 1-point signals (interlace decomposition).
Shuffled input data (bit-reversal). log2N
stages.
(*)(*): only first decomposition shown.
(**)

18/11/2008 84
FFT family tree
Divide & conquerDivide & conquer
N : GCD(N1,N2) <> 1
Ex: N = 2n
Cost: MAPPINGMAPPING.
• Twiddle-factors calculations.
• Easier sub-problems.
• Some algorithms:
Cooley-Tukey,
Decimation-in-time / in-frequency
Radix-2, Radix-4,
Split radix.
N : GCD(*)(N1,N2) = 1
N1, N2 co-prime. Ex: 240 = 16·3·5
Cost: SUBSUB--PROBLEMSPROBLEMS.
• No twiddle-factors calculations.
• Easier mapping (permutations).
• Some algorithms:
Good-Thomas, Kolba,
Parks, Winograd.
(
*)
GCD= Greatest Common Divisor

18/11/2008 85
(Some) FFT concepts
Butterfly: basic FFT calculation element.
s[k+N/2]
s[k]
WN
N/2
WN
0
1. Decimation-in-time ⇒⇒⇒⇒⇒⇒⇒⇒ time data shuffling.
2. Decimation-in-frequency ⇒⇒⇒⇒⇒⇒⇒⇒ frequency data shuffling.
3. In-place computation: no auxiliary storage needed, allowed by most algorithms.
4. DFT pruning: only few bins needed or different from zero ⇒⇒⇒⇒⇒⇒⇒⇒ only they get calculated
(ex: Goertzel algorithm).
5. Real-data case: Mirroring effect in DFT coeffs. ⇒⇒⇒⇒⇒⇒⇒⇒ only half of them calculated.
6. N power-of-two: Many common FFT algorithms work with power-of-two number of
inputs. When they are not ⇒⇒⇒⇒⇒⇒⇒⇒ pad inputs with zeroes.
Dual approach: data to be reordered in
time or in frequency!

18/11/2008 86
Systems spectral analysis (hints)
System analysis: measure inputSystem analysis: measure input--output relationshipoutput relationship..
DIGITAL LTI
SYSTEM
h[n]
x[n] y[n]
H(f) : LTI transfer functionH(f) : LTI transfer function
∑
∞
=
⋅−=∗=
0m
h[m]m]x[nh[n]x[n]y[n]x[n] h[n]
X(f) H(f) Y(f) = X(f) · H(f)
DIGITAL
LTI
SYSTEM
0 n
δδδδ[n]
1
0 n
h[n]
h[t] = impulse response
Linear Time InvariantLinear Time Invariant
y[n] predicted from { x[n], h[t] }
Transfer function can be estimated by Y(f) / X(f)

18/11/2008 87
Estimating H(f) (hints)
(f)XX(f)(f)G *
xx ⋅= Power Spectral Density of x[t] (FT
of autocorrelation).
(f)XY(f)(f)G *
yx ⋅= Cross Power Spectrum of x[t] & y[t] (FT
of cross-correlation).
It is a check on
H(f) validity!
(f)G(f)G
(f)G
(f)C
yyxx
2
yx
xy
⋅
=
Coherence function
- values in [0,1]
- assess degree of linear relationship
between x[t] & y[t].
xx
yx
*
*
G
G
(f)XX(f)
(f)XY(f)
X(f)
Y(f)
H(f) =
⋅
⋅
== Transfer Function
(ex: beam !)

18/11/2008 88
Notes on DFT
The DFT transform is an exact one-to-one transform
The DFT can only approximate the continuous Fourier Transform
The DFT components correspond to N frequencies that are fs/N apart
The DFT of a real-valued signal gives symmetric frequency components
A fast algorithm, the FFT, is available for implementing the DFT
Frequency Resolution of DFT
The frequency resolution of the N-point DFT is
– fr = fs / N
•The DFT can resolve exactly only the frequencies falling exactly at: k * fs/N.
– There is spectral leakage for components falling between the DFT bins
•Zero-padding is often use to provide more resolution in the frequency
components

18/11/2008 89
The Z Transform
Powerful tool for analyzing & designing DT systems
Generalization of the DTFT:
∑ −
n
fnj
enxDTFT π2
][:
fj
rez π2−
=
∑∑ −−−
=
n
n
n
fnjn
znxernxnxZ ][][:]}[{ 2π
∫
−
dtetxCTFT ftj π2
)(:
fjs πσ 2+=
∫∫
−+−
= dtetxdtetxtxL sttfj
)()(:)}({ )2( πσ

18/11/2008 90
- Fourier transform nj
n
j
enxeX ωω −
∞
−∞=
∑= ][)(
limited as it can handle stable systems only.
)( ωj
eX converges, or exists if ∑ ∞<|][| nx
i.e., stable system → Fourier Transform converges
Fourier Transform Limitation
Require to analyze digital system in general

18/11/2008 91
nj
n
nj
r ernxeXLet ωω −
∞
−∞=
−
∑= )][()( converges if ∑ ∞<−
|][| n
rnx
Example:
||
2
1
1
2|2||)(|
|2||)(|
][2][
r
rereX
eeX
nunx
nnnjnnj
r
njnj
n
−
=<=
∞→=
=
∑∑
∑
−−−
−
ωω
ωω
→ converges if 2|| >r
The condition for convergence is relaxed!
FT generalization

18/11/2008 93
In general, if ][][ nuanx n
=
az >ROC is
- In terms of ,)(zX
)( jw
eX is a special case
Where , or1=z 1=r
2
causal
Z Transform
UC: Unit Circle

18/11/2008 94
]1[][ −−−= nuanx n
1
1
1
1
0
1
1
1
1
1
11
1
1
)(1
]1[)(
−
−
−
−
∞
=
−
∞
=
−
−
−∞=
−
∞
−∞=
−
−
=
−
−
=
−
−=
−=−=
−=−−−=
∑∑
∑∑
az
za
za
za
zaza
zaznuazX
n
n
n
nn
n
nn
n
nn
ROC 11
<−
za az <: , or
2=|a|
Example

18/11/2008 95
11
1
0
2
1
1
1
3
1
1
1
)
2
1
()
3
1
()(
−−
−
−∞=
−
∞
=
−
−
−
+
=
−−= ∑∑
zz
zzzX
n
nn
n
nn
]1[)
2
1
(][)
3
1
(][ −−−−= nununx nn
3
1
>z
2
1
<z, 1/21/3
Two Sided Sequence

18/11/2008 96
(1)
(2)
(3)
(4)
(5)
(6)
(7)
][1][ zalln ↔δ
]1[
1
1
][ >
−
↔ z
z
nu
]1[
1
1
]1[ 1
<
−
↔−−− −
z
z
nu
][
1
1
][ 1
az
az
nua n
>
−
↔ −
]0,
,0,0[][
<∞
>↔− −
mifexceptzall
mifexceptzallzmn m
δ
][
1
1
]1[ 1
az
az
nua n
<
−
↔−−− −
][
)1(
][ 21
1
az
az
az
nuna n
>
−
↔ −
−
Z Transform Pairs

18/11/2008 97
(1) in general
(2) absolutely converges
(3) cannot contain a pole
(4) FIR sequence entire z plane, may be except for 0 or
(5) Right-sided sequence outward of the outermost pole
(6) Left-sided sequence inward from the innermost pole
(7) Two-sided sequence a ring in between two adjacent rings
(8) is a connected region
ROC
ROC
)( jw
eX ROCUC ⊂↔
→
→
→
→
∞
ROC properties
∞≤<<≤ LR rROCr0

18/11/2008 98
a b c
(e.g.) If x[n] is a sum of 3 sequences whose poles
are a, b, c respectively,
There exist A possible as shown belowROC
All right-sided All left-sided
two left-sided two right-sided
a b c

18/11/2008 99
(1) Inspection method
(2) Power Series Expansion
(3) Partial Fraction Expansion →
omitted
(e.g.)
power series expansion
∑ ∑
∞
=
∞
−∞=
−
−+
=
−
=
1
1
][
)1(
)(
n n
n
nnn
znx
n
za
zX
=→ ][nx
00
1)1( 1
≤
≥− +
n
n
n
an
n
azazzX >+= −
)1log()( 1
{
Inverse Z Transform
Useful
→

18/11/2008 100

18/11/2008 101
(1) Linearity
(2) Time shifting
)()(][][ 2121 zbXzaXnbxnax +↔+
)(][ zXznnx on
o
−
↔−
Z Transform Properties
(3) Multiplication by an Exponential Sequence
)(][
o
n
o
z
z
Xnxz ↔
(4) Differentiation of X(z)
XRROCzx
dx
d
znnx =−↔ ][][

18/11/2008 102
(5) Conjugation of Complex Sequence
(6) Time-Reversal
(7) Convolution-Integration
XRROCzXnx =↔ )(][ ***
)
1
(][
1
)
1
(][ *
**
z
Xnx
R
ROC
z
Xnx
X
↔−
=↔−
)()(][*][ 2121 zXzXnxnx ↔
(8) Initial Value Theorem
)(lim]0[ zXx
z ∞→
=

18/11/2008 103
(1)
① Causal
② Stable
① Outward
② UC ROC
][)
2
1
(][ nunx n
=
nn
n
zzX −
∞
=
∑= )
2
1
()(
0
1
2
1
1
1
−
−
=
z
⊂
2
1
>z
2
1
1
2
1
1
ROC :
Stability & Causality

18/11/2008 104
(2)
① Anti Causal
② Unstable
① Inward
② UC ROC
]1[)
2
1
(][ −−−= nunx n
1
1
2
1
1
1
)
2
1
()(
−
−
−
−∞=
−
=−= ∑
z
zzX nn
n
2
1
<z
⊄
2
1
2
1
ROC :

18/11/2008 105
(3)
① Causal
② Unstable
ROC :
① Outward
② UC ROC
][)2(][ nunx n
=
1
0 21
1
2)( −
−
∞
= −
== ∑ z
zzX n
n
n
2>z
⊄
1 2
1 2

18/11/2008 106
(4)
① Anti Causal
② Stable
① Inward
② UC ROC
What do you find?
]1[2][ −−−= nunx n
1
1
21
1
2)( −
−
−
−∞= −
=−= ∑ z
zzX n
n
n
2<z
⊂
1 2
ROC :

18/11/2008 107
-Analog systems
(continuous time)
H(s)
st
esH )(st
e
-Digital Systems
(discrete time)
H(z)
n
zzH )(n
z
ττ
ττ
τττ
τ
dehsH
esHdeeh
dtxhx(t)h(t)y(t)
sr
ststs
−
∞
∞−
∞
∞−
−
∞
∞−
∫
∫
∫
⋅=
⋅=⋅⋅⋅=
−⋅=∗=
)()(
)()(
)()(
k
k
nnk
k
k
zkhzH
zzHzzkh
knxkhx(n)h(n)y(n)
−
∞
−∞=
−
∞
−∞=
∞
−∞=
⋅=
⋅=⋅=
−⋅=∗=
∑
∑
∑
)()(
)()(
)()(
Laplace, Z & Fourier

18/11/2008 108
planes − planez −
Ωj
ωj
e
1
1
1
12
−
−
+
−
⋅=
z
z
T
s
d
Laplace transform z-transform
),(),( ππω −∞−∞Ω
LHP inside unit circle
Ω==Ω jssHjH )()( ω
ω
j
ez
j
zHeH =
= )()(
Fouier transforms

18/11/2008 109
<Laplace-Transform> <z-Transform>
Stable : ROC includes Stable : ROC includes
j-axis UC
Causal : ROC Rightward Causal : ROC outward
),(),(
)()(
∞−∞ΩΩ →
=
Ω=
∞
∞−
−
∫
jH
dzezhsH
js
sz
),(),(
][)(
ππωωω
− →
=
=
∞
−∞=
−
∑
jez
n
n
eH
znhzH
j
Analog
LTI
x(t) y(t) Digital
LTI
x[n] y[n]
st
e
st
esH )( n
z n
zzH )(
z-planes-plane

18/11/2008 110
S/Z Plane

18/11/2008 111
DT LTI Systems
CT LTI systems are described by ODEs
∑∑ ==
+=
M
k
k
k
k
N
k
k
k
k tx
dt
d
bty
dt
d
aty
01
)()()(
DT LTI systems are described by Difference Equations
s
ss
T
nTxTnx
dt
tdx )())1(()( −+
≈ • normalised wrt Ts ][]1[
)(
nxnx
dt
tdx
−+≈
Typical DT LTI system Difference Equations are
∑∑ ==
−+−=
M
k
k
N
k
k knxbknyany
01
][][][

18/11/2008 112
DT System Components
DIGITALDIGITAL ANALOGANALOG
(1) Expression
∑∑ ==
−+−=
M
k
k
N
k
k knxbknyany
01
],[][][ ∑∑ ==
+=
M
k
k
k
k
N
k
k
k
k tx
dt
d
bty
dt
d
aty
01
)()()(
(2) Circuit Elements
Adder
Multiplier
Delay
][][ 21 nxnx +
][nxa ⋅
a
]1[ −nx1−
z
R
L
C
VS
CS
iRv ⋅=
dt
di
Lv ⋅=
∫= idt
C
v
1
+
-

18/11/2008 113
DIGITALDIGITAL ANALOGANALOG
Time - Domain
Transform - Domain
Discrete – time Continuous - time
∑
∑
=
−
=
−
−
==
−
N
k
k
k
M
k
k
k
za
zb
zX
zY
zH
zHtransformz
1
0
1
)(
)(
)(
)(,
∑
∑
=
−
=
−
−
==
−
N
k
k
k
M
k
k
k
sa
sb
sX
sY
sH
sHtransformLaplace
1
0
1
)(
)(
)(
)(,
System Transfer Function
Transfer Function

18/11/2008 114
DT System Response
Cz
za
zb
zX
zY
zH N
k
k
k
M
k
k
k
∈∀
−
==
∑
∑
=
−
=
−
1
0
1
)(
)(
)(
∑∑ ==
−+−=
N
i
i
N
i
i inhainbnh
10
][][][ δ
•z = r.ej2πf are the eigenvectors of the Difference Equation
•The Magnitude of the system response at each z is given by the
•System Transfer Function
•We resolve the i/p sequence x[n] on these eigenvectors z (ZT)
•Find the response at each Z (apply xfer func)
•Sum the responses back to get o/p (IZT)
z3
z1
z2
z4
a1
a2b1
b2
System i/p
System o/p?
b20
b1o
a2o
a1o

18/11/2008 115
Convolution Sum
So given a system and i/p, we do
][)(
)()()(
)(][
nyzY
zXzHzY
zXnx
⇒
=
⇒
System
•IZT
∑
∞
−∞=
−=
k
knxkhny ][][][
•Convolution
•Impulse
response

18/11/2008 116
Causality & Stability
an LTI system is causal if and
only if its impulse response is
zero for negative values of n
– i.e. system o/p does not depend
on future values of i/p
LTI system is Stable if its impulse
response is absolutely summable
∑
∞
=
−=
0
][][][
k
knxkhny
∞<∑
∞
−∞=
|][|
k
kh

18/11/2008 117
Schur - Cohn Stability Test
Write the denominator of system function H(z) as
Convert the polynomial coefficients to reflection
coefficients
– (given in next section: lattice filters)
H(z) is a stable system if
Conversion can be done using Levinson-Durbin Algorithm
N
NN zazazazA −−−
++++= ......1)( 2
2
1
1
ka
mK
mKm ∀<1||

18/11/2008 118
Discrete Cosine Transform
Go 2 Top

18/11/2008 119
Discrete Cosine Transform
DCT uses Cosines as its basis (N point sequence)
Each Left/Right Boundary can be Even or Odd.
– 2 choices per boundary
Symmetry about a data point or the point halfway between
two data points.
– 2 choices per boundary.
Total of 16 possibilities
Sequences with Even left boundary, form 8 types of DCTs.
Sequences with Odd left boundary, form 8 types of DSTs.
119

18/11/2008 120
Boundary Symmetries
120

18/11/2008 121
Energy Compaction
The boundary conditions are responsible for the Energy
Compaction Property of the DCT.
Boundaries affect the Rate of Convergence.
Any discontinuity in function reduce the rate of convergence
Which leads to more sinusoids in the function representation
The Rate of Convergence is that
– how fast a sequence converges to its limit point.
Single Principle for Signal Compression
– Smoother the function, the faster will be the convergence
– Lesser the terms required for representing the function
– Higher is the compression achieved
121

18/11/2008 122
Audio frame input

18/11/2008 123
DFT has discontinuties at the boundaries,
– means reduced rate of convergence.
DCTs have smooth boundaries
N point DFT has N bins over a span of Fs
N point DCT has N bins over a span of Fs/2.
The odd left boundary in DST implies a discontinuity
Hence DCT gives better compression than DFT and DST
For non-stationary, high transients signals such as
– percussion sounds in audio
– high frequency components in images
– DCT is not that much efficient
DWT gives better compression for such signals
DCT DST DFT DWT

18/11/2008 124
DFT of audio frame

18/11/2008 125
DST of audio frame

18/11/2008 126
Four main types of DCTs
126

18/11/2008 127
DCT of audio frame

18/11/2008 128
IDCT Kernels
The inverse DCT kernels of various types of DCTs are:
128

18/11/2008 129
Blocking Artifacts
During block transform coding in images, audio and video.
– Quantizing the spectral coefficients results in discontinuity
at block boundaries.
– Leading to appearance of high frequency content.
– These are called blocking artifacts
Low Pass filtering is one easy solution,
– It results in blurring/smearing

18/11/2008 130
Preventing Blocking Artifacts
Overlapping & Windowing of data blocks
Overlap/Add Process requires doubling of data length,
– which increases 100% redundancy.
Time Domain Aliasing in a special way will help us
remove this 100% redundancy.
MDCT/TDAC is the answer to all these problems.

18/11/2008 131
Modified DCT: MDCT
The main idea is to create a lapped transform
– To avoid the artifacts arising due to block boundaries
– To obtain a good Energy compression like DCTs.
Artifacts like Ringing, Pre-echo, drop-outs, warbling, metallic
ringing, hissing etc. may arise due to Copression Schemes.
MDCT is employed in MP3, AC3, Ogg Vorbis, AAC, WMA for
audio Compression.
131

18/11/2008 132
MDCT Cont…..
MDCT is based on DCT-IV and maps 2N real numbers
to N real numbers
It is designed for consecutive blocks of data where
second half of first block overlaps with first half of second
block.
otherwise it is not invertible during IMDCT process. ?????
132
∑
−
=
−=+++=
12
0
1,.....,1,0for)]
2
1
)(
22
1
([
N
n
nk
Nkk
N
n
N
CosxX
π
n
x
kX

18/11/2008 133
DCT-IV,MDCT and TDA
For an MDCT of a 2N point sequence (a,b,c,d), where each
block is of size N/2.
– +N/2 phase term in MDCT signifies a shift of N/2 in the input
The MDCT of 2N inputs (a,b,c,d) is exactly equivalent to
DCT-IV of N inputs
The inverse DCT-IV will simply give back the sequence
It is clearly Time Domain Aliasing: TDA
133
),( RR
badc −−−
),( RR
badc −−−
OrderReversedinputbeenhasthat xmeansxR

18/11/2008 134
∑
−
=
−=+++=
1
0
12,.....,1,0for)]
2
1
)(
22
1
([
1 N
n
kn Nnk
N
n
N
CosX
N
y
π
2/),,,( RRRR cddcabba ++−−
2/),,,( RRRR effecddc ++−−
MDCT/IMDCT of a single block is not invertible
Because of Time Domain Aliasing after IMDCT,
– Due to 2N to N point mapping during MDCT…
Remember Undersampling ???
IMDCT {MDCT(a,b,c,d)} =
IMDCT {MDCT(c,d,e,f)} =
Where IMDCT is given by

18/11/2008 135
MDCT Cont…..
Perfect invertibility is achieved by adding the overlapped
IMDCTs of the consecutive overlapping blocks
50% Overlap Add will Result into Time Domain Aliasing
Cancellation: TDAC
135
TDAC:onCancellatiAliasingDomainTimeasKnownisThis
:),(
2/),,,(
2/),,,(
dc
effecddc
cddcabba
RRRR
RRRR
=
++−−
+
++−−

18/11/2008 136
Windowing
Windowing is done in order to smoothen the boundaries of
the block to reduce the discontinuties to zero at n=0 and 2N
for improved transformation.
A Good Window will have lesser sidelobe energy hence a
good estimation of frequency component.
The transform still remains invertible, i.e. TDAC works, if a
symmetric window satisfies the Princen-Bradley condition i.e.
136
1
22
=+ + Nnn ww

18/11/2008 137
MDCT as Filter Bank
Window is the prototype FIR filter
Different windows can be designed.
The N output coeff. of a single 2N->N point transform
can be assumed to be output of N Sub band filter
banks.
Transform of the continuous 2N data point frames
with N point overlap can be interpreted as convolution
with output decimated by N.
More on this in the next topic.

18/11/2008 138
MDCT as Filter Bank

18/11/2008 139
Digital Filter Structures
Go 2 Top

18/11/2008 140
Finite Impulse Response
∑=
−=
M
k
knxkhny
0
][][][
∑=
−=
M
k
k knxbny
0
][][
Finite Impulse Response
FIR Filters
Non-Recursive
Convolution Based Implementation
)2()1()()(: 210 −+−+= nxbnxbnxbnyFIR
The impulse response vanish
after finite number of samples

18/11/2008 141
Infinite Impulse Response
∑
∞
=
−=
0
][][][
k
knxkhny
∑∑ ==
−+−=
M
k
k
N
k
k knxbknyany
01
][][][
Infinite Impulse Response
IIR Filters
Recursive
Feedback Based Implementation
)1()()(: −+= naynxnyIIR
The impulse response does not
vanish after finite number of samples
z-1
a
x[n] y[n]
+

18/11/2008 142
][*][][ nhnxny = )()()( zHzXzY =
)(
|)(|)(
ω
ωω j
eHjjj
eeHeH ∠
⋅=
(e.q) Frequency selective filter - ideal
|)(| ωj
eH
cωcω− π ω
c
j
eH ωωω
<= ||,1|)(|
elsewhere,0
αωω
−=∠ )( j
eH
)(
)(sin
][
απ
αω
−
−
=
n
n
nh c
( : delay, centerpoint of sync function)α
1
Frequency Response
)( ωj
eH∠
π ω
α−

18/11/2008 143
Group delay
)(arg)(grd)( ωω
ω
ωτ jj
eH
d
d
eH −==
αωω
−=)(arg).( j
eHqe αωτ =)(
Generalized Linear Phase System
ωαωω jj
R
j
eeHeH −
⋅= )()(
PartReal
delayconstant)(
phaselinear)(arg
←=
←−=
α
αω
ω
ω
j
j
eHgrd
eH
Group Delay

18/11/2008 144
)orderhighestthehalf(
2
M
=αphase!linear)(
...})1
2
(cos2
2
cos2{
...)()(
...)(
)Cond.Sufficient(,...)(
10
2
)1
2
()1
2
(
2
1
222
0
0
)1(
110
1
10
→⋅=
+−+=
++++=
++++=
=+++=
−
−
−−−−−−
−−−−
−
−−
ωωα
ω
ωωωωωω
ωωωω
ωω
j
R
j
M
j
M
j
M
j
M
j
M
j
M
j
M
j
MjMjjj
nMn
M
M
eHe
M
a
M
ae
eeeaeeea
eaeaeaaeH
aazazaazH
Linear Phase
LTI System
90 degree
phase shift
50 Hz i/p wave
100 Hz i/p wave
50 Hz o/p wave
90 degree
Phase shift
100 Hz o/p wave
180 degree
Phase shift
180 degree
phase shift

18/11/2008 145
∑∑ ==
−+−=
M
k
k
N
k
k knxbknyany
01
][][][
∑
∑
=
−
=
−
−
= N
k
k
k
M
k
k
k
za
zb
zH
1
0
1
)(
(1) Direct Form-I
)(nx )(ny
1−
z
1−
z
1−
z
1−
z
1−
z
1−
z
)1( −nx
)1( +− Mnx
)( Mnx −
0b
1b
1−Mb
Mb )( Nny −
)1( +− Nny
)1( −ny
1a
1−Na
Na
IIR filter Structure

18/11/2008 146
ordercascadingoftindependen:SystemLTI→
)(nx )(ny
1−
z
1−
z
1−
z
1−
z
1−
z
0b
1b
Mb
1a
1−Na
Na
)duplicated(
delaysyunnecessarRemove→
1−
z
1−
z
1−
z
0b
1b
Mb
Na
1a
)(nx )(ny
Direct Form-II
Canonical Form
Require Minimum Memory Elements

18/11/2008 147
- Second order factored form
- pole-zero pairing
 
 
∏
∏
∑
∑
=
−−
=
−−
=
−
=
−
−−
++
⋅=
−
= 2/
1
2
2
1
1
2/
1
2
2
1
1
0
1
0
)1(
)1(
1
)( N
k
kk
M
k
kk
N
k
k
k
M
k
k
k
zz
zz
b
za
zb
zH
αα
ββ
L
11α
21α
12α
22α
L1α
L2α L2β
L1β11β
21β 22β
12β
1 11
Cascade Form

18/11/2008 148
- Partial fraction expansion
∑ ∑∑ −
−−
−
−
+
−−
+
+
−
= k
k
kk
kk
k
k
k
zC
zz
zBB
z
A
zH 2
2
1
1
1
10
11
)(
ββα
0C
1C
k2β
k1β
kB0
kB1
kA
kα
Parallel Form

18/11/2008 149
- Reverse the flow of a structure,
then you will get the identical transfer function
1a
1a
2a
0b
1b
2b
1−
z
1−
z
1−
z
1−
z
1a
2a
0b
1b
2b
)(nx )(nx
)(ny )(ny
true?itisWhy:Question formulasMason'→
Transposed Form

18/11/2008 150
(1) Direct Form
Z-1 Z-1 Z-1
)(nx
)0(h )1(h )2(h
)1( −nx )2( −nx
)1( +− Nnx
)1( −Nh
)(ny
.....
⊕
∑
−
=
−=
1
0
)()()(
N
k
knxkhny
∑
−
=
−
=
1
0
)()(
N
k
k
zkhzH
FIR Filter Structures

18/11/2008 151
(2) Cascade Form
)()( 2
0
1
0
2/
1
0
−−
=
++= ∏ zzzH kk
N
k
k δδδ
⊕ ⊕ ⊕....)(nx 00δ )(ny
10δ
20δ
01δ
11δ
21δ
(3) Parallel Form (Frequency Sampling)
−
=
−
−
==
1
0
2
)(IDFT)(
1
)(
N
k
N
j
N
kn
N eWWkH
N
nh
π
Cascade Form

18/11/2008 152
n
N
k
N
k
kn
N zWkH
N
zH −
−
=
−
=
−
∑ ∑= ))(
1
()(
1
0
1
0
n
N
k
N
k
k
N zWkH
N
)()(
1 1
1
0
1
0
−
−
=
−
=
−
∑ ∑= 1
1
0 1
1
)(
1
−−
−−
= −
−
= ∑ zW
z
kH
N k
N
NN
k
1
1
0 1
)(1
−−
−
=
−
−
−
= ∑ zW
kH
N
z
k
N
N
k
N
+
N
z−
+
+
+
1−
z
1−
z
.
.
.
0
NW
1+− N
NW
)1( −NH
)0(H
)(ny
N
1
-1
)(nx
Parallel Form

18/11/2008 153
)()( nMhnh −=
)(nx
)(
)0(
Mh
h =
)1(
)1(
−
=
Mh
h
)(ny
oddM :
evenM :
)(
)0(
Mh
h =
)(nx
)(ny
Linear Phase FIR Structure

18/11/2008 154
Lattice Filter Structure
The lattice filter is extensively used in digital speech
processing and in implementation of adaptive filter.
It is a preferred form of realization over other FIR or
IIR filter structures because in speech analysis and
in speech synthesis the small number of coefficients
allows a large number of formants to be modeled in
real-time.
– All-zeros lattice is the FIR filter representation of the lattice filter.
– The lattice ladder is the IIR filter representation.

18/11/2008 155
All-zero Lattice Filters
An FIR filter of order M has a lattice structure with M stages.
x(n)
f0(n)
g0(n)
K0
1/z 1/z 1/z
g1(n) gM-1(n)g2(n) gM(n)
f1(n) f2(n) fM-1(n) fM(n)
y(n)
K1
K1
K2
K2
KM
KM
MmngnfKng
MmngKnfnf
nxngnf
mmmm
mmmm
,,2,1),1()()(
,,2,1),1()()(
)()()(
11
11
00
L
L
=−+=
=−+=
==
−−
−−
Km: reflection coefficients

18/11/2008 156
All-zero lattice filters
∑ ∑= =
−−






+==
M
m
M
m
mmm
m z
b
b
bzbzH
0 1 0
0 1)(
If the FIR filter is given by the direct form
And if we denote the polynomial
Mm
b
b
mzmzA m
M
M
m
m
MM ,,1,)(;)(1)(
01
L==





+= ∑=
−
αα
Then the lattice filter coefficients {Km} can be obtained by the
following recursive algorithm

18/11/2008 157
1,,1);(
1,,;
1
)()(
)(
1,,);()(
)(
21
1
L
L
L
−==
=
−
−
=
==
=
−
−−
MmmK
Mm
K
zBKzA
zA
MmzAzzB
MK
mm
m
mmm
m
m
m
m
MM
α
α
Note that the above algorithm will fail if |Km|=1 for any m.
Clearly, this condition is satisfied by the linear-phase FIR filter.
Therefore, linear-phase FIR filter cannot be implemented using lattice structure.
This algorithm can also be used for finding reflection coefficients of All Pole lattice
IIR filter and also test the stability using Schur - Cohn criteria
Reflection Coefficients

18/11/2008 158
All-pole Lattice Filter
A lattice structure for an IIR filter is restricted to an all-pole
system unction.
It can be developed from an FIR lattice structure.
An IIR filter of order N has a lattice structure with N stages
Each stage of the filter has an input and output that are related
by the order-recursive equations.
)(
1
)(1
1
)(
1
zA
zma
zH
N
N
m
m
N
=
+
=
∑=
−

18/11/2008 159
All-pole Lattice Filter
x(n)=fN(n)
gN(n)
1/z 1/z
g1(n)g2(n) g0(n)
fN-1(n) f2(n) f0(n)
y(n)
KN
-KN
K2
-K2 -K1
K1
gN-1(n)
f1(n)
1/z
)()()(
1,,1,),1()()(
,1,),1()()(
)()(
00
11
11
ngnfny
NNmngnfKng
NNmngKnfnf
nxnf
mmmm
mmmm
N
==
−=−+=
−=−−=
=
−−
−−
L
L

18/11/2008 160
Lattice ladder Filters
A general IIR filter containing both poles and zeros can be
realized as a lattice-type structure by using an all-pole lattice
as the basic building block.
Consider an IIR filter with system function
Where, without loss of generality, we assume that N>=M
)(
)(
)(1
)(
)(
0
0
zA
zB
zka
zkb
zH
N
M
N
k
k
N
M
k
k
M
=
+
=
∑
∑
=
−
=
−

18/11/2008 161
Basic 2nd Order IIR Structures
Direct form I realization
5 Multiply 4 Additions per y(n)
4 registers storing x(n-1), x(n-2),
y(n-1), y(n-2)
Direct form II realization aka BiQuad
5 multiply, 4 additions per y(n)
2 registers storing w(n-1), w(n-2)
b(0)
b(1)
b(2)
x(n)
−a(1)
−a(2)
u(n) y(n) x(n)
−a(1)
−a(2)
w(n) y(n)b(0)
b(1)
b(2)
z-1
z-1
z-1
z-1
z-1
z-1
x(n-1)
x(n-2)
y(n-1)
y(n-2)
w(n-1)
w(n-2)
2
2
1
1
2
2
1
10
1
)( −−
−−
−−
++
=
zaza
zbzbb
zH

18/11/2008 162
Applications of Biquad Digital Filters
The general comb+biquad structure can be used to implement numerous digital signal
processing functions:
Moving average, differentiater, integrator, leaky-integrator, 1st and 2nd order delay network,
Goertzel network, sliding DFT, real oscillator, quadrature oscillator, comb filter
Bandpass filter, equalizer, real frequency sampling filter, DC bias removal, etc.
z-1
z-1
z-N
y(n)x(n)
c1
a2 b2
b1
b0
a1
a0
Comb 2nd-order recursive network (biquad)
-+
2
2
1
1
2
2
1
10
1
1
)1()( −−
−−
−
−−
++
−=
zaza
zbzbb
zczH N

18/11/2008 163
Effects of pole-clustering
∑
∑
=
−
=
−
−
= N
k
k
k
M
k
k
k
za
zb
zH
0
0
1
)(
- after quantization
)(ˆ)(
ˆ
ˆ
zHzH
bbbb
aaaa
kkkk
kkkk
→
∆+=→
∆+=→
- sensitivity of (in denominator) on H(z) is larger
than that of :
kaˆ
kbˆ
Change poles changed denominator of H(z) changedka
Coefficient Quantization
Finite Word Length issues:
1) Coefficient Quantization effects
2) Round Off Noise in multiplication
3) Overflow in addition
4) Limit Cycles

18/11/2008 164
Let ∑ Π=
−
=
−
−=−=
N
k
k
N
k
k
k zpzazP
1
1
1
)1(1)(
kk aa ˆ→ kkkk pppp ∆+=→ ˆ
Then
k
i
pizipizk a
p
p
zP
a
zP
∂
∂
⋅
∂
∂
=
∂
∂
==
)()(
k
i
iik
N
ik
k
k
i
a
p
pppp
∂
∂
⋅−−=− −−
≠
=
−
Π
11
1
)1(
)()1(
1
1
1
1
ki
N
ik
k
kN
i
ik
N
ik
k
k
i
k
i
pp
p
pp
p
a
p
−
=
−
=
∂
∂
ΠΠ
≠
=
−
−
≠
=
+−
Therefore, a small change in could cause a large change in
when is close to , (or when poles are clustered), causing
a large change in H(z) or its frequency spectrum,
Even it can destabilize an IIR filter having pole near unit circle
ka ip
kpip
To reduce the coefficient
quantization effects
higher order filters should
be implemented as cascade
or parallel structures of
second order filter sections
Called Biquads
Moreover pole zero pairing of
each biquad section
can be done to reduce
Output round-off noise power
Harmonics distortion due to
round-off effects in
multiplications can be reduced
by dithering

18/11/2008 165
Y(n) = ay(n-1) - x(n) -> y(n) = Q[ay(n-1)] - x(n)
x[n] y[n]
z-1
x[n]
y[n]
z-1Q[]
a
a
Limit Cycle
Magnitude truncation can be used
to eliminate limit cycles
Eventually decays Oscillations
Dead
Band

18/11/2008 166
Comparison
Linear Phase design possible
– Constant group delay
Always stable
Robust to quantization errors
– e.g. rounding error
Large number of taps required
– High computational load
Non linear phase
– Delay varies with frequency
Can be unstable
Sensitive to quantization errors
– e.g. limit cycle
– Harmonic distortion
Small number of taps required
– Low computational load
Direct Form-II requires less memory elements
Cascade Form IIR filter structure is one of least sensitive to quantization
Lattice & Parallel Direct Form-II show small sensitivity to coefficient quantization
Cascade and parallel Direct Form-I require minimum intermediate word length
Comparison of various structures
FIR filters IIR filters

Dsp fundamentals part-ii

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