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ECE 468 Digital Signal Processing
1. History:• Digital signal processing has its roots in 17th and 18th
century mathematics.
• The techniques and applications of this field are as old as Newton and Gauss and as new as digital computers and integrated circuits
2. Definition:• Digital signal processing (DSP) is concerned with the
representation of the signals by a sequence of numbers or symbols and the processing of these signals.
• Digital signal processing and analog signal processing are subfields of signal processing.
3. Applications:• DSP includes subfields like:
Audio and speech signal processing, sonar and radar signal processing, sensor array processing, spectral estimation, statistical signal processing, digital image processing, signal processing for communications, biomedical signal processing, seismic data processing, etc.
• DSP is not confined to 1D signals. Sometimes 2D, 3D or 4D signals.
• Until recently, signal processing has typically been carried out using analog equipment. With the development of computers, more and more DSP applications. It was often useful to simulate a signal processing system on a computer before implementing it in analog hardware.
1
Broadcasting: television and radio programs.
Sound waves digital signals broadcasted/receivedanalogous format and filtered
Telecommunications: transfer signals, etc. If satellites are usedAudio waves electromagnetic waves wireless mediumAudio waves light waves transfer by optical fibres …
Navigation: Devices or systems such as SONAR or Radar work primarily on the basis of DSP. For example, SONAR makes use of sound waves (signals) in order to calculate the depth. On the other hand, radars make use of radio waves in order to communicate thelocations of various objects in a particular radius.
2
Radar remote sensingTo investigate the Earth and solar system using radar remote sensing techniques. DSP for geoscience applications.
www.stanford.edu/group/radar/
Biomedical Applications: DSP is used extensively in the field of biomedicine. In it, the various signals that are generated by the different organs in the human body are measured in order to findinformation regarding the health of the same. For example, in case of electrocardiograms (ECG), the electric signals generated by the heart are measured.
3
Digital imaging:
sp.cs.tut.fi/
Apart from those mentioned above, digital signal processing has various other applications. For example, it is used in cars,remote controls, seismic analysis etc. Thus, DSP proves to be one of the most useful techniques developed in the modern times.
4
Chapter 1: Discrete-Time Signals and Systems
Signal and System:• A signal can be defined as a function that conveys
information, generally about the state or behavior of a physical system.
• In the fields of communications, signal processing, and in electrical engineering, a signal is any time-varying or spatial-varying quantity.
• In DSP, engineers usually study digital signals in one of the following domains: time domain, spatial domain, frequency domain, etc.
What is time domain and spatial domain?
For example: Speech signal in a time domain; picture in a spatial domain
Time (seconds)
magnitude
1D
Pixel locations (mm)
Pixel loc
2D
• Signals are represented mathematically as functions of one or more independent variables.
• The independent variable of the mathematical representation of a signal may be either continuous or discrete.
• Continuous-time signals: signals that are defined at a continuum of times and thus are represented by continuous variable functions.
• Discrete-time signals: defined at discrete times and thus the independent variable take on only discrete values; i.e., discrete-signals are represented as sequences of number.
• Our example: Speech signals and pictures may have either a continuous or a discrete variable representation, and if certainconditions hold, these representation are entirely equivalent.
•Digital signals are those for which both time and amplitude are discrete.
•Continuous-time, continuous-amplitude signals are sometimes called analog signals.
• Signal processing:
transformoriginal signal --> another signal(Voice to light, light to light, sound to electrical signals, etc…Or separate 2 signals, enhance some components of signals, etc…
• Continuous-time system: both input and output are continuous-time signals;
• Discrete-time system: both input and output are discrete-time signals.
•Digital systems Input and output are digital signals;
• DSP deal with transformation of signals that are discrete in both amplitude and time.
5
• Continuous-time system: both input and output are continuous-time signals;
• Discrete-time system: both input and output are discrete-time signals.
•Digital systems Input and output are digital signals;
• DSP deal with transformation of signals that are discrete in both amplitude and time.
•Advantages of DSP: flexibility using computers or with digital hardware; to simulate analog systems or to realize signal transformations impossible to realize with analog hardware.
1.1 Discrete-Time Signals –Sequences • Focus on signals represented by sequences.
• x(n): sequence of number x, the nth number in the sequence is denoted x(n)
X(0)
The sequence: -1, -2, 1, 2, 3, 2, 1, -1X(0)=3;X(1)=2;X(-3)=??
Some special defined sequences:1. Unit-sample sequence: (discrete-time impulse=impulse)
2. Unit step
⎩⎨⎧
=≠
=0,10,0
)(nn
nδ
⎩⎨⎧
<≥
=0,00,1
)(nn
nu
⎩⎨⎧
=≠
=0,10,0
)(nn
nδ⎩⎨⎧
<≥
=0,00,1
)(nn
nu
)1()()(
)()(
−−=
= ∑−∞=
nunun
knun
k
δ
δ
Draw a u(n) and u(n-1) ??
3. Real exponential
…….. ……..
Any sequence whose values are of theForm an where a is a real number.
6
4. Sinusoidal
A sinusoidal sequence has values of the form:
)cos( 0 Φ+nA ω
Periodic:
If x(n)=x(n+N) for all n;N is the period.
Energy: 2
)(∑∞
−∞=
=n
nxε
X(0)What’s the energy for this sequence?
• Sequence calculation: )()( nynxyx ⋅=⋅Product:
)()( nynxyx +=+Sum:
Multiplication: )(nxx ⋅=⋅ αα
0
x(n): -2,-1,3,2,0,1
0
n=0
y(n): 1,0,2,1,2,3,1
n=0
What is the Product, sum, and 2 times ofx(n)?
An arbitrary sequence can be expressed as a sum of scaled, delayedunit samples.
0
Delay or Shift: )()( 0nnxny −= n and n0 are integers
0)(nx )2( +nx0)2( −nx
0
)3()1()()1()2()( 31012 −+−+++++⋅= −− nanananananx δδδδδ
What is a-1 in this case?
∑∞
−∞=
−=k
knkxnx )()()( δ
Generally, an arbitrary sequence:
1.2 Linear Shift-Invariant Systems
• System: a unique transformation or operator that maps an input sequence x(n) into an output sequence y(n).
y(n)=T[x(n)] T: transformation
T[ ] y(n)x(n)
• Linear system: relatively easy to characterize mathematically,used widely for modeling. Non-linear system: complicate.
• Linear system: defined by the principle of superposition. If:y(n)=T[x(n)]
Then a system is linear if and only if:
)()()]([)]([)]()([ 212121 nbynaynxbTnxaTnbxnaxT +=+=+
for any arbitrary constants a and b.
Is y(n)=2 x(n) a linear system?
Is y(n)=[x(n)]2 a linear system???
)()()]()([)]()([ 212
2121 nbynaynbxnaxnbxnaxT +≠+=+
7
•Cannot have exponents (or powers)For example, x squared or x 2
•Cannot multiply or divide each otherFor example: xy; x/y
•Cannot be found under a root sign or square root sign (sqrt)For example: sqrt (x)
Linear expressions:
x + 42x + 4
2x + 4y
Not linear expressions:
X2
2xy+42x/ySqrt(x)
∑∞
−∞=
−=k
knkxnx )()()( δ
Generally, an arbitrary sequence:
y(n)=T[x(n)]
)()()( ∑∞
−∞=
−=k
knkxTny δ
∑∑∑∞
∞−
∞
∞−
∞
−∞=
−⋅=−⋅=−= )()()()()()()( knhkxknTkxknkxTnyk
δδ
Shift-invariant: y(n) x(n); y(n-k) x(n-k)
For linear shift-invariant system:
)()]([ knhknT −=−δ
h(n) is the unit-sample response / impulse response)()( nhnx ∗
• Convolution:
)(*)()()()( nxnhnhnxny =∗=
The convolution order doesn’t matter.
h1(n) y(n)x(n)
y(n)x(n)
h1(n)*h2(n) y(n)x(n)
h2(n)
h2(n) h1(n)
Above three are same for a linear shift-invariant system.What are the impulse responses of above three?
h1(n)+h2(n) y(n)x(n)
h1(n)y(n)x(n)
h2(n)
Above two are same expressions for a linear shift-invariant system.What is the impulse response?
Example:
)(20,00,2
)( nunn
nh nn
=⎩⎨⎧
<≥
= )4()()( −−= nununx
Let’s calculate y(n):
( )∑∞
−∞=
−==∗=k
knhkxnxnhnhnxny )()(*)()()()(
0
u(n) u(n-4)
x(n)=u(n)-u(n-4)
…………
0
…… ……
4
0 3
Convolution:
⎩⎨⎧
<−≥−
=−−
0,00,2
)(kn
knknh
kn
h(1-k)=21-k (1>=k) h(2-k)=22-k (2>=k)
00…… ……1
2
4
8
…… ……
4
8
2
0 3
x(k)
h(0-k)=20-k (0>=k)
0…… ……
12
4
8
8
( )∑∞
−∞=
−==∗=k
knhkxnxnhnhnxny )()(*)()()()(
( ) 3012)1()1( =++=−= ∑∞
−∞=kkhkxy
( ) 7124)2()2( =++=−= ∑∞
−∞=kkhkxy
( ) 15148)3()3( =++=−= ∑∞
−∞=k
khkxy
……
( ) 1001)0()0( =++=−= ∑∞
−∞=kkhkxy
……
Make sure you understand how to calculate the convolution.
Convolution:( )∑
∞
−∞=
−==∗=k
knhkxnxnhnhnxny )()(*)()()()(
)(20,00,2
)( nunn
nh nn
=⎩⎨⎧
<≥
= )4()()( −−= nununx
Easy way to do convolution:
0
…… ……1
2
4
8
flip
0 3
0…… ……
12
4
8
0 3
0…… ……
12
4
8
y(0)=1
0 3
…… ……1
2
4
8
y(2)=1+2+4=7
0 3
…… ……1
2
4
8
y(1)=2+1=3
y(3)=15
0 3
…… ……1
2
4
8
0 3
0…… ……
12
4
8
y(-1)=0
Let’s shift to the left for negative n:
⎩⎨⎧
<≥
=0,0
0,2)(
nnn
nh )5()()( −−= nununx
Let’s calculate:
What is y(-1),y(0),y(1)?
y(0)=00…… ……
2
4
8
6
0 4
0
…… ……2
4
8
6
h(n)=2n n>=0
flip
0 4
0
…… ……2
4
8
6
y(1)=2
0 4
0
…… ……2
4
8
6
y(-1)=0
Let’s calculate textbook problems 2.(a):
0
12
1
x(n) h(n)
0
)()()( nhnxny ∗=Draw your y(n)?
9
1.5 Frequency-domain representation of discrete-time systems and signals
• representations of signals in terms of sinusoids or complex exponentials (i.e. Fourier representations)
Rectangular Function magnitude of the Fourier transform
Suppose the input sequence is: njenx ω=)( ∞<<∞− n
A complex exponential of radian frequency ω
∑
∑∑∞
−∞=
−
∞
−∞=
−−∞
−∞=
=
===
k
kjj
jnj
k
kjnjknj
k
ekheHhere
eHeekheekhny
ωω
ωωωωω
)()(:
)()()()( )(
Frequency response: H(ej ω) describes the change in complex amplitude of a complex exponential as a function of the frequency ω.
)()()( ωωω jI
jR
j ejHeHeH +=
Example: a system with unit-sample response:
elsewhereNn
nh10
01
)(−≤≤
⎩⎨⎧
=
ω
ωωω
j
NjN
n
njj
eeeeH −
−−
=
−
−−
== ∑ 11)(
1
0
ωωω sincos je j +=N=5
Nπ2
Nπ2
− π π2π2−
Magnitude of the frequency response H(ejω) of the system With unit-sample response
• For a general sequence x(n), we define the Fourier transform as:
∑∞
−∞=
−=n
njj enxeX ωω )()(
Inverse Fourier transform:
ωπ
ωωπ
πdeeXnx njj )(
21)( ∫−=
• If x(n) is absolutely summable: ∞<∑∞
−∞=n
nx |)(|
The series x(n) is absolutely convergent, the frequency responsefor a stable system will always converge.
• Ideal low-pass filter
ππ− coωcoω− π2π2−
1|)(| ωjeH
⎩⎨⎧
≤<≤
=πωω
ωωω
||0||1
)(co
cojeH
ωπ
ωωπ
πdeeXnx njj )(
21)( ∫−=
Let’s calculate the impulse response h(n):
nnnj
jn
ejn
edjn
dedeeHnh
coco
co
conjnj
njnjj
co
co
co
co
πωω
π
ωω
ππ
ωπ
ωπ
ωω
ω
ω
ω
ω
ωωωπ
π
)sin()sin(22
12
1)(2
1
121)(
21)(
==
−⋅==
⋅==
∫
∫∫
−
−−
1/2
π1
π1
π31
−π31
−
π51
π51
0
If 2πω =co
10
1.6 Symmetry Prosperities of the Fourier Transform
• For a general sequence x(n), we define the Fourier transform as:
∑∞
−∞=
−=n
njj enxeX ωω )()(
Inverse Fourier transform:
ωπ
ωωπ
πdeeXnx njj )(
21)( ∫−=
• Even (symetric): a real-valued sequence x(n) is called even, if xe(-n)=xe (n)
• Odd (antisymetric): a real-valued sequence x(n) is called even, if xo(-n)=-xo (n)
X(0) X(0)
even odd
• Even (symmetric): xe(-n)=xe (n)
• Odd (antisymmetric): xo(-n)=-xo (n)
For an arbitrary real-valued sequence x(n):
)]()([21)(
)]()([21)(
)()()(
nxnxnx
nxnxnx
wherenxnxnx
o
e
oe
−−=
−+=
+=
Test if xe(n) is even and xo(n) is odd???
• Conjugate Symmetric: xe*(-n)=xe (n)
x(1)=a+bj; x(-1)=a-bj;
• Conjugate Antisymmetric: xo*(-n)=-xo (n)
x(1)=a+bj; x(-1)=-a+bj;
For an arbitrary sequence x(n):
)](*)([21)(
)](*)([21)(
)()()(
nxnxnx
nxnxnx
wherenxnxnx
o
e
oe
−−=
−+=
+=
For complex-valued sequence:
Test if xe(n) is even and xo(n) is odd???
A Fourier Transform X(ejω) can be expressed as:
)](*)([21)(
)](*)([21)(
)()()(
ωωω
ωωω
ωωω
jjjo
jjje
jo
je
j
eXeXex
eXeXex
whereeXeXeX
−
−
−=
+=
+=
)(*)( ωω je
je eXeX −=Conjugate Symmetric:
)(*)( ωω jo
jo eXeX −−=Conjugate Antisymmetric:
1.7 Sampling of Continuous-Time Signals
• Continuous-time, continuous-amplitude signals are sometimes called analog signals. Example: voltage, current, temperature,…
• Digital signals: discrete both in time and amplitude–Example: attendance of this class, digitizes analog signals,…
• Discrete signals are often derived from continuous-time signals by periodic sampling.
• Understand how the derived sequence is related to the originalsignal
11
Periodic Sampling
• Sampling is a continuous to discrete-time conversion
• Most common sampling is periodic• T is the sampling period in second• fs = 1/T is the sampling frequency in Hz• Sampling frequency in radian-per-second Ωs=2πfs
rad/sec
[ ] ( ) ∞<<∞−= nnTxnx a
[ ] ( ) ∞<<∞−= nnTxnx aLet’s give an example:
t
xa(t)=sin(t)
0 π2π2π
If sampling period T=4π
.......
1)2
sin()4
2sin()2(
)4
sin()4
1sin()1(
0)4
0sin()0(
)4
sin()4
()()(
==⋅=
=⋅=
=⋅=
⋅=⋅==
ππ
ππ
π
ππ
x
x
x
nnxnTxnx aa
0 4
x(n)
3n
21
• Fundamental issue in digital signal processing– If we lose information during sampling, we cannot recover it
• Under certain conditions an analog signal can be sampled without loss so that it can be reconstructed perfectly
0-1
2 4 6 8 10
0
1
∫
∫∞
∞−
Ω−
+∞
∞−
Ω
=Ω
ΩΩ=
dtetxjX
dejXtx
tjaa
tjaa
)()(
)(2
1)(π
Fourier representation of an analog signal xa(t):
∫
∫
−
+∞
∞−
Ω
=
ΩΩ==
π
π
ωω ωπ
π
deeXnx
dejXnTxnx
njj
nTjaa
)(21)(
)(21)()(Sampling:
Discrete FT:
Finally, we can have:
)2(1)(T
rjjXT
eXr
aTj π
+Ω= ∑∞
−∞=
Ω )2(1)(T
rjT
jXT
eXr
aj πωω += ∑
∞
−∞=
)2(1)(T
rjjXT
eXr
aTj π
+Ω= ∑∞
−∞=
Ω )2(1)(T
rjT
jXT
eXr
aj πωω += ∑
∞
−∞=
Ω
Xa(jΩ)
0
1
0Ω
X(ejΩT))]2([
TjX a
π−Ω⋅
Tπ2
)]2([T
jX aπ
+Ω⋅
Tπ2
−
T1
)2(1)(T
rjjXT
eXr
aTj π
+Ω= ∑∞
−∞=
Ω
T/ω=Ω
0 4
x(n)
3n
21
So, if you are doing sampling in the time/space domain:
0Ω
X(ejΩT))]2([
TjX a
π−Ω⋅
Tπ2
)]2([T
jX aπ
+Ω⋅
Tπ2
−
T1
Your Fourier performance:
Can cut off to go back xa(t)
12
( )ΩjXs
Ωs 2Ωs 3Ωs-2Ωs Ωs3Ωs
Ωs<Ω0
( )ΩjXa
20Ω
−2
0Ω Ωs 2Ωs 3Ωs-2Ωs Ωs3Ωs
Ωs>Ω0)( ωjeX
20Ω
−2
0Ω
20Ω
−2
0Ω
1
1/T
1/T
aliasing
Aliasing: The phenomenon, where in effect a high-frequency Component in Xa(jΩ) takes on the identity of a lower frequency.
0Ω
X(ejΩT))]2([
TjX a
π−Ω⋅
Tπ2
)]2([T
jX aπ
+Ω⋅
Tπ2
−
T1
Ω0/2
Tπ2
0 <Ω Can go back to the original xa(t)
We sample at a rate at least twice the highest frequency of Xa(jΩ), Then X(ejw) is identical to Xa(ω/T) in the interval πωπ ≤≤−
0ω
X(ejω)
T1 T/ω=Ω)]2([ πω −⋅jX a)]2([ πω +⋅jX a
π2π2−
1.8 Two-dimensional sequences and systems
• 2D unit-sample sequence:
• 2D unit-step sequence:
• 2D exponential sequence:
• 2D sinusoidal sequence:
• Separable sequence:
⎩⎨⎧ ≥≥
=otherwise
nmnmu
,00,0,1
),(
)cos()cos( 10 θωω +Φ+⋅ nmA
⎩⎨⎧
==≠
=0,10,,0
),(nm
nmnmδ
nmba
)()(),( 21 nxmxnmx ⋅=
Above unit-sample, unit-step, exponential, sinusoidal sequences separable?
)cos(),( 0 nmnmx ⋅= ω not separable.
• An arbitrary 2D sequence can be expressed as a linear combinationof shifted unit sample:
∑ ∑∞
−∞=
∞
−∞=
−−⋅=k r
rnkmrkxnmx ),(),(),( δ
∑∞
−∞=
−=k
knkxnx )()()( δRecap: 1D sequence:
• For a linear shift-invariant system:
∑ ∑
∑ ∑
∑ ∑
∞
−∞=
∞
−∞=
∞
−∞=
∞
−∞=
∞
−∞=
∞
−∞=
−−=
−−⋅=
−−⋅=
=
k r
k r
k r
rnkmhrkx
rnkmTrkx
rnkmrkxT
nmxTnmy
),(),(
),(),(
),(),(
)],([),(
δ
δ
Convolution
The convolution order doesn’t matter.
• Stable system: a bounded input a bounded output2D linear shift-invariant systems are stable if and only if
• Casual system: 2D linear shift-invariant systems are casual if and only if
h(m,n)=0 for (m<0, n<0).
∞<= ∑∑∞
−∞=
∞
−∞=
Δ
rkrkhS ),(
• 2D Fourier Transform & Inverse Fourier Transform:
2121
2
21
21
21
),(4
1),(
),(),(
ωωπ
π
π
π
π
ωωωω
ωωωω
ddeeeeXnmx
eenmxeeX
njmjjj
n
njmj
m
jj
∫ ∫
∑∑
− −
∞
−∞=
−−∞
−∞=
=
⋅⋅=
13
Z-transform: a generalization of the Fourier Transform
• For a sequence x(n):
∑∞
−∞=
−⋅=n
nznxzX )()(
• Denote the z-transform as: )]([ nxℑ
Z is a complex variable
• If express the complex variable z in polar form as:
∑∑∑ −−−∞
−∞=
− ⋅⋅==⋅= njnnj
n
n ernxrenxznxzX ωω )())[()()(
If r=1: z-transform Fourier transform
1|| ==⋅=
zeerz jj ωω
Z-transform properties:
1) Shift of a sequence:
+−
+−
<<=+ℑ
<<=ℑ
xxn
xx
RzRzXznnx
thenRzRzXnx
||),()]([
||),()]([
00
***0)0(
0 )()()0()]([ nm
m
nm
m
n
n
zzmxzmxznnxnnx −∞
−∞=
−−∞
−∞=
−∞
−∞=∑∑∑ ⋅=⋅=⋅+=+ℑ
2) Convolution of sequences:
)()()(
)(*)()(
zYzXzWthen
nynxnw
=
=
14
Let’s demonstrate it:
)()(
])(][)([])([)(
])([)(
)()()()()()(
)(*)()(
)(
zYzX
zmyzkxzzmykx
knmzmykx
zknykxzknykxznwzW
nynxnw
k
m
n
k
k
km
n
k
km
m
k
n
n
n
n k
n
n
=
⋅=⋅=
−=⋅=
−⋅=−==
=
∑ ∑∑ ∑
∑ ∑
∑ ∑∑ ∑∑
∞
−∞=
−∞
−∞=
−∞
−∞=
−−∞
−∞=
∞
−∞=
+−∞
−∞=
∞
−∞=
−∞
−∞=
−∞
−∞=
∞
−∞=
−∞
−∞=
Convergence region: satisfy both
+−
+−
<<<<
yy
xx
RzRRzR
||||
Periodic Sequences Discrete Fourier Series
• Consider a sequence periodic with period N:)(~)(~ kNnxnx += for any integer value of k
∑
∑−
=
−
=
−
⋅=
⋅=
1
0
)/2(
1
0
)/2(
)(~1)(~
)(~)(~
N
k
nkNj
N
n
nkNj
ekXN
nx
enxkX
π
π
∑
∑−
=
−
−
=
−
⋅=
⋅=
=
1
0
1
0
)/2(
)(~1)(~
)(~)(~
N
k
nkN
N
n
nkN
NjN
WkXN
nx
WnxkX
eW π
We can also express as:
• are periodic sequences.)(~ kX)(~ nx
15
Let’s see if is periodic:)(~ kX
∑−
=
−
⋅=
=+=1
0
)/2(
)(~)(~),(~)(~
N
n
nkN
NjN
WnxkX
eWkNnxnx π
We know is periodic:)(~ nx
)(~)]2sin()2[cos()(~)(~
][][)(~
][)(~
][)(~
)(~)(~
2
)/2(1
0
)/2(
1
0
)/2(
1
0
)()/2(
1
0
)(
kX
njnkXekX
eenx
enx
enx
WnxNkX
nj
nNNjN
n
nkNj
N
n
nNnkNj
N
n
NknNj
N
n
NknN
=
−⋅=⋅=
⋅=
⋅=
⋅=
⋅=+
−
−−
=
−
−
=
+−
−
=
+−
−
=
+
∑
∑
∑
∑
πππ
ππ
π
π
Let x(n) represent one period of )(~ nx
elsewherenxNnfornxnx
0)(10)(~)(
=−≤≤=
…….…….
)(~ nx
N0-Nx(n)
N0
∑−
=
−=1
0)()(
N
n
nznxzX
∑−
=
⋅=1
0)(~)(~ N
n
nkNWnxkX
kN
kNj WezzXkX −==
= )/2()|()(~π
16
kN
kNj WezzXkX −=== )/2()|()(~
π
z-plane
ω
ωjezzX =)(
Nπ2
z-plane
kNjezkX )/2()(~ π=
The Discrete Fourier Series corresponds to sampling the z-transformat N points equally spaced in angle around the unit circle.
)(~ kX
)( zX
Look at above figures, what is N in this case??? N=12
Illustration: let’s consider a sequence:
)10/sin()2/sin(
1)(~)(~
)10/4(
4
0
)10/2(
4
010
1
0
kke
e
WWnxkX
kj
n
nkj
n
nkN
n
nkN
πππ
π
−
=
−
=
−
=
=
=
⋅=⋅=
∑
∑∑
…….…….
)(~ nx
40-10 10
N=101
17
kN
kNj WezzXkX −=== )/2()|()(~
π
)2
sin(
)2
5sin(
)10/5sin(
)2/5sin()(
52
10)/2(
2
)10/5
4(
ω
ωπωπ
πωπ
πω
πω
πω
ω
πω
πω
j
jj
e
eeX
k
kN
−
−
=
==
====>
=
Discrete Fourier Series z transform(periodic ) (finite x(n) ))(~ nx
)(nx
40-10 10
1
• Properties of the Discrete Fourier Series:
Linearity:
)(~)(~)(~)(~)(~)(~
213
213
kXbkXakX
nxbnxanx
⋅+⋅=
⋅+⋅=
)(~1 nx )(~
2 nx periodic, both with period N
All sequences are periodic with period N
kXbkXa
enxbenxa
enxbnxa
enxkX
nkNjN
n
nkNj
N
n
nkNj
N
n
nkNj
(~)(~
])(~)(~[
)](~)(~[
)(~)(~
21
)/2(2
1
0
)/2(1
1
0
)/2(21
1
0
)/2(33
⋅+⋅=
⋅⋅+⋅⋅=
⋅⋅+⋅=
⋅=
−−
=
−
−
=
−
−
=
−
∑
∑
∑
ππ
π
πLet’s see:
18
Periodic Convolution:
)(~1 nx )(~
2 nx periodic, both with period N )(~)(~??? 21 kXkX ⋅==>
∑∑
∑∑
∑∑
∑∑
−
=
+−
=
−
=
−
=
−
=
−
=
−
−
=
−
=
−
⋅⋅=
⋅⋅=⋅
⋅=⋅=
⋅=⋅=
1
0
)(21
1
0
1
021
1
021
1
02
1
0
)/2(22
1
01
1
0
)/2(11
)(~)(~
)(~)(~)(~)(~
)(~)(~)(~
)(~)(~)(~
N
r
krmN
N
m
N
r
rkN
mkN
N
m
N
r
rkN
N
m
rkNj
N
m
mkN
N
m
mkNj
Wrxmx
WWrxmxkXkX
WrxerxkX
WmxemxkX
π
π
]1)[(~)(~
)(~)(~1
)(~)(~1)(~
)(1
0
1
021
1
0
1
0
)(21
1
0
1
0
21
1
03
krmnN
N
k
N
r
N
m
N
r
krmN
N
m
nkN
N
k
nkN
N
k
WN
rxmx
WrxmxWN
kXkXWN
nx
−−−−
=
−
=
−
=
−
=
+−
=
−−
=
−−
=
∑∑∑
∑∑∑
∑
⋅=
⋅⋅=
⋅=
DFS
Sequence:
⎩⎨⎧ ⋅+−=
=−−−−
=∑ otherwise
NlmnrW
Nkrmn
N
N
k 0)(11 )(
1
0
l is any integer.
∑
∑∑
∑∑∑
−
=
−
=
−
=
−−−−
=
−
=
−
=
−⋅=
⋅=
⋅=
1
021
1
021
1
0
)(1
0
1
021
1
03
)(~)(~
)(~)(~
]1)[(~)(~)(~
N
m
N
r
N
m
krmnN
N
k
N
r
N
m
mnxmx
rxmx
WN
rxmxnx
Periodic Convolution
∑∑−
=
−
=
−⋅=−⋅1
012
1
021 )(~)(~)(~)(~ N
m
N
m
mnxmxmnxmxOrder doesn’t matter
19
∑
∑−
=
−
=
−⋅==>⋅
⋅==>−⋅
1
02111
21
1
021
)(~)(~1)(~)(~
)(~)(~)(~)(~
N
l
N
m
lkXkXN
nxnx
kXkXmnxmx
Periodic convolution productProduct 1/N times the periodic convolution of )(~
1 kX )(2~ kX
)(~2 mx
0 N)(~
1 mx
)(~2 mx −
)2(~2 mx −
-N
)2(~)(~21 mxmx −⋅
20
Finite-Duration Sequences Discrete Fourier Transform (DFT)
• Rectangular sequence
⎩⎨⎧ −≤≤
=otherwise
NnnRN 0
101)(
∑∞
−∞=
+=r
rNnxnx )()(~ )()(~)( nRnxnx N=
Discrete Fourier Transform (DFT):
⎪⎩
⎪⎨
⎧−≤≤⋅
=
⎪⎩
⎪⎨
⎧−≤≤⋅
=
=
∑
∑
−
=
−
−
=
−
otherwise
NnWkXNnx
otherwise
NkWnxkX
eW
N
k
nkN
N
n
nkN
NjN
0
10)(1)(~
0
10)()(
1
0
1
0
)/2( π
)(nx
0 N)(~ nx
)2(~ +nx
Circular Shift of a sequence
)())2(()(1 nRnxnx NN+=
……
……
……
……
x1(n) is not a linear shift of x(n). It is a circular shift.
21
• Image a finite-duration sequence x(n) displayed around the circumference of a cylinder (circumference of N points)
)())2(()())(()(~)(~
1
1
nRnxnxmnxmnxnx
NN
N
+=+=+=
⎩⎨⎧ −≤≤
=otherwise
NnnRN 0
101)(
)(~)(1 kXWkX kmN−=
DFT:
Symmetry properties of DFT: similar to previous DFS.
Periodic convolution DFS product
We already learned:
)(~)(~)(~)(~21
1
021 kXkXmnxmx
N
m⋅==>−⋅∑
−
=
• Let’s look at finite-duration sequences x1(n) and x2(n), duration NDFT: X1(k) and X2(k) X1(k)X2(k)??
x3(n) is one period of ∑−
=
−⋅=1
0213 )(~)(~)(~ N
mmnxmxnx
)()(
)(]))(())(([
)(])(~)(~[)(
21
1
021
1
0213
nxnx
nRmnxmx
nRmnxmxnx
NN
N
mN
N
N
m
=
−⋅=
−⋅=
∑
∑−
=
−
=
N
22
0)(:
101
00)()(
1
0
0
0
01
knNWkXDFT
Nnnnn
nnnnnx
=
⎪⎩
⎪⎨
⎧
−≤<=
<≤=
−= δExample 1:
)()( 21 nxnx NLet’s solve ???
0 N
)(nx
)(2 mx
)(1 mx
n0=1
0 N
)(~2 mx −
…………
)())0((2 mRmx NN−
x3(0)
)())1((2 mRmx NN−
…………)1(~
2 mx −
x3(1))()( 21 nxnx Nx3(n)=
23
Example 2:
⎩⎨⎧ −≤≤
==otherwise
Nnnxnx
0101
)()( 21
0 N
)(1 nx
1
)(2 nx1
)(3 nxN
Linear convolution using the DFT
∑−
=
−=1
0213 )()()(
N
mmnxmxnx
Linear convolution: N-point sequences: x1(n) x2(n)
The length of x3(n) is 2N-1
For length of 2N-1:
)()]()([12
1)(
)()(
)()(
12
22
012213
22
012212
22
01211
nRWkXkXN
nx
WnxkX
WnxkX
N
N
k
nkN
N
n
nkN
N
n
nkN
−
−
=
−−
−
=−
−
=−
⋅⋅−
=
⋅=
⋅=
∑
∑
∑
∑−
=
−=1
0213 )()()(
N
mmnxmxnx is the linear convolution
24
Two-dimensional DFT:
• Rectangular sequence
⎩⎨⎧ −≤≤−≤≤
=otherwise
NnMmnmR NM 0
10,101),(,
),(),(1),(
),(),(),(
,
1
0
ln1
0
,
1
0
ln1
0
nmRWWlkXMN
nmx
lkRWWnmxlkX
NM
N
lN
kmM
M
k
NM
N
nN
kmM
M
m
⋅⋅=
⋅⋅=
∑∑
∑∑−
=
−−−
=
−
=
−
=
If x(m,n) is separable, then the 2D DFT is also separable. Linearity:
),(),(),(),(),(),(
213
213
lkbXlkaXlkXnmbxnmaxnmx
+=+=
Flow Graph and Matrix Representation of Digital Filters
4.1 Signal flow graph representation of digital networks
• Basic block-diagram symbols:
+)(1 nx
)(2 nx
)()( 21 nxnx +
a)(nx )(nax
Addition of two sequences
multiplication
Z-1)(nx )1( −nx Unit delay
Shift property of z-transform:Z-transform of x(n-1) is simply z-1 times the z-transform of x(n)
25
)()2()1()( 21 nbxnyanyany +−+−=
+)(nx b
+
Z-1
a1
Z-1a2
)(ny
)1( −ny
)2( −ny
Digital network
For Digital hardware:We must provide storage for y(n-1) and y(n-2) and also the constants a1, a2, and b, as well as means for multiplication andaddition.
Node j, Wj
Node k, Wk
• Source node: no entering branches.
• The node value at a source node j will be denoted as xj
• The output of a branch connecting source node j to network node k will be denoted as sjk.
Node k
Source node jxj
26
+
Z-1
+y(n)x(n)
a b
Delay branchSink node
Source node
a by(n)x(n)
1 2 3
4
N=4 network nodes;M=1 source nodes.
jk
N
nodesnetwork
jk
jk
M
nodessource
jjk
N
nodesnetwork
jk
ry
svw
∑
∑∑
=
==
=
+=
)(
1
)(
1
)(
1
w1(n)=s11(n)+v41(n)w2(n)=v12(n)w3(n)=v23(n)+v43(n)w4(n)=v24(n)y(n)=w3(n)
For example:
)([
0001
)1()1()1()1(
0010000000000000
)()()()(
000000
0001000
)()()()(
4
3
2
1
4
3
2
1
10
1
4
3
2
1
nx
nwnwnwnw
nwnwnwnw
bb
a
nwnwnwnw
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
[ ] )(
)()()()(
0100)( 3
4
3
2
1
nw
nwnwnwnw
ny =
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
Delay branch y(n)x(n)w1 w2
w3
w4
a1 b1
b0
27
cω
|H(ejw)|
cωπ −2 π2Frequency response of an ideal lowpass filter
"template" used for the specification of a model lowpass filter in the frequency domain dd
IIR filter design methods:
-- Other computer-aided techniques, etc
A.
28
transform an analog filter design to a digital filter design:--- choose the unit-sample response of the digital filter as equallyspaced samples of the impulse response of the analog filter
)()( nThnh a= T: sampling period
)()( nThnh a=
Impulse invariance mapping:
Mapping Example:
29
Impulse invariance design example:
)1)(1()cos(1
12/1
12/1)(
2/12/1)(
)(
11
1
11
22
−−−−−
−−
−−−−−
−−−
=
−+
−=
−++
++=
+++
=
zeezeezbTe
zeezeezH
jbasjbas
basassH
jbTaTjbTaT
aT
jbTaTjbTaT
a
If:
∑= −
=N
k k
ka ss
AsH1
)( ∑=
−−=
N
kTs
k
zeAzH
k1
11)(
30
Impulse response truncation: to obtain a finite-length impulse response by truncating an infinite-duration impulse response sequence.
∫
∑
−
∞
−∞=
−
=
=
π
π
ωω
ωω
ωπ
deeHnh
enheH
njjdd
n
njd
jd
)(21)(
)()(
Ideal desired frequency response:
Truncation:
⎩⎨⎧ −≤≤
=otherwise
Nnnhnh d
010)(
)(
B.
1.
31
Computer-aided design of FIR filters– Frequency Sampling
• In the Fourier-Window design method: The desired frequency response of the FIR filter is given in thecontinuous frequency; then use a window to deduce the filter.
• In the frequency sampling method:The desired frequency response of the filter is given at discrete frequencies which are the uniform samples of the continuous frequency response, then the inverse DFT is used to obtainthe corresponding discrete-time impulse response.
It allows us to design frequency selective filters, leads to FIR filters whose coefficients are integers, making the computation very fast even less precise.
32
Chapter 6 FFT:
Decimation-in-Time FFT algorithms• To decompose the DFT computation into successively smallerDFT computations.
• Decimation-in-time algorithms: decomposition is based on decomposing the sequence x(n) into successively smaller subsequences. • The principle of decimation-in-time is most conveniently illustrated by considering the special case of N an integer power of 2;
vN 2=• Since N is an even integer, we can consider computing X(k) by separating x(n) into two N/2-point sequences consisting of the even-numbered points in x(n) and the odd-numbered points in x(n). With X(k) given by
1,.....,1,0)()(1
0−== ∑
−
=
NkWnxkXN
n
nkN
An 8-point sequence, i.e. N=8:
N/2 pointDFT
N/2 pointDFT
x(0)
x(2)
x(4)
x(6)
x(1)
x(3)
x(5)
x(7)
G(0)
G(1)
G(2)
G(3)
H(0)
H(1)
H(2)
H(3)
X(4)WN
4
X(0)WN
0
X(1)WN
1
X(2)WN
2
X(3)WN
3
X(5)WN
5
X(6)WN
6
X(7)WN
7
33
The efficient FFT algorithm in the computational flow graph representation for N=8 is obtained as shown below:
34
Decimation-in-frequency FFT algorithm:
• The decimation-in-time FFT algorithms were all based upon the decomposition of the DFT computation by forming smaller and smaller subsequences.
• Alternatively decimation-in-frequency FFT algorithms are all based upon decomposition of the DFT computation over X(k). For N, a power of 2:
• We divide the input sequence into first half and the last half of points so that:
signal flow graph for the case of 8-point DFT
35
Proceeding in a manner similar to that followed in deriving the decimation-in-time algorithm, the final signal flow graph for computation is shown as
36
Recommended