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Principles of Telecommunications
• Communication is the largest sector of the
electronics field, hence knowledge &
understanding is a must for every student
• The field of Electronic Communication changes
so fast
• Need for Firm Grounding in Fundamentals: also
understanding of the real world components,
circuits, equipment & systems in everyday use.
• Balance in Principles of latest techniques
• Study the system level understanding
EEB317 Principles of Telecoms
•Signals & Systems
•Amplitude Modulation
•Angle Modulation
•Detection & Demodulation
•Noise in Communications System
Signals & Systems
•Classification
•Power & Energy Spectral Density
•Correlation
•Fourier Series & Transform
•Convolution
Overview
• Tx of information between 2 distant points
• Dominated by 4 important sources: speech,
television, facsimile & personal computers
• Three basic processes:
• Transmitter, Channel & Receiver
Communication System
Communication System
Eeb317 principles of telecoms 2015
Classification of signals & Systems
• A system is an interacting set of physical
objects or physical conditions called system
components
• A signal: set of information or data. Can be
input, output or internal
• Signals may be functions of independent
variables such as time, distance, force, position,
pressure, temp … for simplicity only time will be
used in this class
• Mathematical models are mathematical
equations that represent signals & systems
• They permit quantitative analysis and design of
signals & systems
• Continuous-time signal x(t), has a value
specified for all points in time, & a continuous-
time system operates on and produces
continuous-time signals
Continuous-time signal
Signals
Signals
• Discrete-time signal: Signal specified only at
discrete values
• Analog signal: Signal whose amplitude can
take on any value in a continuous range
• Digital signal: Signal whose amplitude can
take on only finite number of values (M-ary)
• Periodic signal: Signal g(t) is periodic if for
some +ve constant T0 (period):
)()( 0Ttgtg 
DT signals
Energy & Power Signals
• Energy signal: Signal with finite energy & zero
power
• Power signal: Signal with finite power & infinite
energy
 


dttgE
2
)(












2/
2/
2
2/
2/
2
)(
1
lim
)(
1
lim00
T
TT
T
TT
dttg
T
P
dttg
T
P
Are these energy or power signals?
•
Energy & Power Signals examples
• Since in (a) amplitude approaches zero
• It’s an energy signal:
• Since (b) is periodic, it’s a power signal:
t
 





84422)(
0
20
1
22 2/
dtedtdttgE
t
3
1
3
2
1
)(
1
)(
1
lim
32
1
1
1
1
1
21
1
22/
2/
2


 
t
dttdttg
T
dttg
T
P
T
TT
Worked example
• Determine the power of the following signals
)cos()(   tCtg












2/
2/
22
2/
2/
22
2/
2/
22/
2/
2
2
)]22cos(1[
2
)(cos
1
)cos(
1
)(
1
lim
T
T
T
T
T
T
T
TT
C
dtt
T
C
P
dttC
T
dttC
T
dttg
T
P


Work these out
• (b)
• (c)
)cos()cos()( 222111   tCtCtg
21  
tj
Detg 0
)( 

More examples
• Compute the signal energy & signal power for
the following complex valued signal and
indicate whether the signal is an energy or
power signal
tj
Aetg 2
)( 
More examples
• Since g(t) is a periodic signal, it cannot be an
energy signal. Therefore compute the signal
power first. Signal Period:
• Since the signal has a finite power, it is a power
signal & has infinite energy (VERIFY!)

1
0 T
  2/122
2
2 1
1
/11
1
/11
1
AtAdtAdtAeP
t
t
tj
t
t
t
t







Deterministic & Random Signals
• Deterministic signal: Physical description is
known in either a mathematical or graphical
form. eg:
• Random signal: Signal known only in terms of
probabilistic description such as mean value,
mean square value rather than its complete
mathematical or graphical description
• eg: noise signal, message signal
)1(tan)( 1
 
ttg
Deterministic & Random Signals
• Use matlab to plot a deterministic signal:
• & random noise
• To use matlab you must:
• (a) declare the variables
• (b) since it’s first time, use the plot command
• (c) label the plot
)1(tan)( 1
 
ttg
Signal Operations
• Time shifting: If a signal g(t) is time shifted by
t1 units, it is denoted as f(t) =g(t-t1).
• If t1>0, the shift is to the right (time delay)
• If t1<0, the shift is to the left (time advance)
• To demonstrate time shifting plot the signals:
)1(tan)(
)1(tan)(
)(tan)(
1
1
1






ttY
ttG
ttg
Time Shift matlab code
• close all; % close graghs
• clear all; % clear all the variables & functions from memory
• t = -5:.3:5; % declear variable "t"
• g = atan(t);
• G = atan(t-1);
• Y = atan(t+1);
• plot(t,g); % plots the function "g"
• hold;
• plot(t,G,'r'); % plots “G”
• plot(t,Y,'k');
• hold off;
• grid on;
• xlabel('t');
• ylabel('g(t),G(t) & Y(t)');
• title('time shifting demonstration');
• legend('g-original','G-delay','Y-advance');
Time Shift gragh
Signal Operations
• Time Scaling: Compression or expansion of a
signal
• Signal f(t) is g(t) compressed by a factor of ‘a’ if
f(t) = g(at), therefore f(t/a) = g(t) for a>1
• Similarly f(t) is g(t) expanded (slowed down) by
a factor of ‘a’ if f(t) = g(t/a), therefore f(at) = g(t)
for a<1
• To time-scale a signal by a factor of ‘a’, replace
t with at.
• If a > 1 the scaling is compressed & if a < 1,
the scaling is expanded.
Time Scaling demonstration
• close all;
• clear all;
• t = -5:.3:5;
• f = sawtooth(t);
• G = sawtooth(2*t);
• F = sawtooth(t/2);
• plot(t,f); hold;
• plot(t,G,'r');
• grid on;
• xlabel('t');
• ylabel('f(t) & G(t)');
• title('time scaling demonstration');
• legend('f','G');
Time Scale gragh
Signal Operations
• Time Reversal/Inversion/Folding:
• To time reverse a signal, replace t with –t
• If f(t) is a time resersal of g(t) then
• f(t)=g(-t)
• See the matlab code of g(-t)
Time Reversal Demo-code
• close all; clear all;
• t = -5:.3:5;
• g = atan(t);
• G = atan(-t);
• plot(t,g); % hold;
• plot(t,G,'r'); % hold off;
• grid on;
• xlabel('t');
• ylabel('g(t) & G(t)');
• title('time Reversal Demonstration');
• legend('f-original', 'G-timeReversed');
Eeb317 principles of telecoms 2015
• Continous time signal: g(t)
• Samples of continuous-time signal: g(nT)
• Discrete-time signal: g(n)
Samples of Continuous-time signal
• close all; clear all;
• t=-5:.5:5;
• g=atan(t);
• plot(t,g);
• ylabel('g(t)');
• grid on;
• title('Continuous-time signal Demonstration');
• figure
• stem(t,g);
• grid on;
• xlabel('nT');
• ylabel('g(nT)');
• title('Samples of Continuous-time signal');
Eeb317 principles of telecoms 2015
Delta Function
• Delta/Dirac/Unit Impulse function:
Rectangular pulse with an infinitesimally small
width & infinitely large height & an overall area
of unity.





1)(
0)(
dtt
t

 0t
Multiplying a function by delta
• Sampling/Sifting property of delta
  )()(()
)(*)0()()(
TtTfTttf
tfttf




)()()()()(
)0()()0()()(
TfdtTtTfdtTttf
fdttfdtttf


 
 










Sampling/sifting property
• The area under the product of a function with
delta is equal to the value of that function at the
instant where delta is located
• Function f(t) must be continuous where the
delta is located
Unit Step Function
 




t
d
1
0
)(  0
0


t
t


t
tud )()( 
)(t
dt
du

Time Shifted, scaled, reversed step
• Causal function: It’s zero before t = 0 otherwise
is non-causal









)(
)(
)(
a
b
tu
a
b
tu
batu
0
0


t
t
Correlation
• Correlation coefficient cn:
• Cross-correlation:
• Autocorrelation:
11  nc



 dttztg
EE
c
zg
n )()(
1



 dttztggz )()()( 



 dttgtgg )()()( 
Fourier Series
• Fourier analysis considers signals to be
constructed from a sum of complex
exponentials with appropriate frequencies,
amplitude & phases
• Frequency components are the complex
exponentials (sines & cosines) which, when
added together, make up the signal
• Orthogonality of signal set: ntxtxtx )(),...(),( 21
Generalized Fourier Series
nm
nm







1
2211
)()(
...)(...)()()(
n
nn
nn
txctg
txctxctxctg
21 ttt 
dtxtg
Edtx
dtxtg
c
t
t
n
n
t
t
n
t
t
n
n 

 


2
1
2
1
2
1
)(
1)(
2





n
t
t
nm
E
dtxtx
0
)(
2
1
Fourier coefficients
nm xx , Are othorgonal
Exponential FS
• Orthogonality:
• Expon. FS



 

0
)( 0
)(
0
00
0
0
T
dtedtee
T
tnmjtjn
T
tjm 
nm
nm









0
0
0
)(
1
)(
0
T
tjn
n
n
tjn
n
dtetg
T
D
eDtg


Parseval’s Theorem
• Energy of the sum of orthogonal signals is
equal to the sum of their energies:
• Parseval’s theorem:
1
2
1
2
1
2
1
2
1
2
1
222
111
2
1
2
1
)()(
)....()(
);()(
EcdttxcdttxcE
txctg
txctg
t
t
t
t



1g
n
n
ng
g
EcE
EcEcE


2
2
2
21
2
1 ...
Trigonometric Fourier Series
• Trig. FS

















01
1
01
1
01
1
01
1
01
1
0
0
0
0
0
0
0
2
0
0
1
00
2sin)(
2
)(
1
2cos)(
2
2cos
2cos)(
2sin2cosag(t)
Tt
t
n
Tt
t
Tt
tTt
t
Tt
t
n
n
n
n
tdtfntg
T
b
dttg
T
a
tdtfntg
Ttdtfn
tdtfntg
a
tfnbtfna





Fourier Transform & Spectra of
Aperiodic Signals
• The spectrum of a periodic signal is found from
FS of a signal over one period
• Since the FS is a periodic function of time, it is
equal to an aperiodic signal only over the FS
expansion interval, outside this interval it
repeats
• FS is used to produce the spectrum of the
periodic extension but not the spectrum of
aperiodic signal
• To find the spectrum of an aperiodic signal we
use FT
Fourier Transform (FT)
• To develop the FT, let’s start with the exponential FS
representation of a periodic signal over the interval
-T/2<t<T/2
• Let the interval increase until the entire time axis is
encompassed
• Since FT is developed from the FS the conditions for
the existence follow from those of the Dirichlet
conditions



dttg )(
Eeb317 principles of telecoms 2015
Fourier Transform
)()(
)]([)()(
)]([)()(
)(
1
)(
12
2
2/
2/
0
0
0
fGtg
fGdfefGtg
tgdtetgfG
dtetg
T
D
eDtg
ftj
ftj
T
T
tjn
n
n
tjn
n























Fourier Transform (Spectrum of g(t))
Inverse Fourier Transform
Fourier Transform pair
Fourier Transform Theorems
• FT characteristics are expressed in the form of
theorems
• The theorems are useful in computing FT of
complicated signals
• Linearity:
• If x(t) X(f) & y(t) Y(f)
• Then
• ax(t)+by(t) aX(f)+bY(f)
• Integral in a linear operation



Fourier Transform Theorems
• Time Shift
• If
Then
Prove:
02
0 )()(
)()(
ftj
efXttx
fXtx



)(22)(2
0
)(22)(2
0
00
2
00
00
00
)(])([)]([
)()()]([
)()]([
tfjfjtfj
tfjfjtfj
ftj
efXdexettx
deexdexttx
ttddttt
dtettxttx


























Fourier Transform Theorems
• Time Reversal
)()(
)()(
fXtx
fXtx


Fourier Transform Theorems
• Time scale (Scale change)
)/(
1
)(
)()(
afX
a
atx
fXtx


Frequency Translation
• If
• Then
• Modulation
• If
• then
)()(
)()(
0
2 0
ffXetx
fXtx
tfj



)(
2
1
)(
2
1
2cos)(
)()(
000 ffXffXtftx
fXtx



Inverse FT
• Find the inverse FT of the delta function
)(2
)(2
2
1
)(
2
1
)(
2
1
)]([
)]([)()(
0
0
0
00
1
12
0
0
0
0



























tj
tj
tj
tjtj
ftj
e
e
e
ede
fGdfefGtg
FT of cosine
• Find the FT of the sinusoid (cosine)
)()(][cos
)](2)(2[
2
1
)](
2
1
[][cos
)(
2
1
cos
000
000
0
00
00










t
eet
eet
tjtj
tjtj
FT of rectangle
• Find the FT of a rectangle of width tau













  
  



/22/0)2/(sin
)2/(sin)(
)2/(sin)2/sin(
2/
)2/sin(
2
)(
2
)(2
)(
1
)(
)()(
2/2/
2/2/
2/
2/
nnc
c
t
rect
cX
j
ee
ee
j
X
dtedte
t
rectX
jj
jj
tjtj














Eeb317 principles of telecoms 2015
Eeb317 principles of telecoms 2015

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Eeb317 principles of telecoms 2015

  • 1. Principles of Telecommunications • Communication is the largest sector of the electronics field, hence knowledge & understanding is a must for every student • The field of Electronic Communication changes so fast • Need for Firm Grounding in Fundamentals: also understanding of the real world components, circuits, equipment & systems in everyday use. • Balance in Principles of latest techniques • Study the system level understanding
  • 2. EEB317 Principles of Telecoms •Signals & Systems •Amplitude Modulation •Angle Modulation •Detection & Demodulation •Noise in Communications System
  • 3. Signals & Systems •Classification •Power & Energy Spectral Density •Correlation •Fourier Series & Transform •Convolution
  • 4. Overview • Tx of information between 2 distant points • Dominated by 4 important sources: speech, television, facsimile & personal computers • Three basic processes: • Transmitter, Channel & Receiver
  • 8. Classification of signals & Systems • A system is an interacting set of physical objects or physical conditions called system components • A signal: set of information or data. Can be input, output or internal • Signals may be functions of independent variables such as time, distance, force, position, pressure, temp … for simplicity only time will be used in this class
  • 9. • Mathematical models are mathematical equations that represent signals & systems • They permit quantitative analysis and design of signals & systems • Continuous-time signal x(t), has a value specified for all points in time, & a continuous- time system operates on and produces continuous-time signals
  • 12. Signals • Discrete-time signal: Signal specified only at discrete values • Analog signal: Signal whose amplitude can take on any value in a continuous range • Digital signal: Signal whose amplitude can take on only finite number of values (M-ary) • Periodic signal: Signal g(t) is periodic if for some +ve constant T0 (period): )()( 0Ttgtg 
  • 14. Energy & Power Signals • Energy signal: Signal with finite energy & zero power • Power signal: Signal with finite power & infinite energy     dttgE 2 )(             2/ 2/ 2 2/ 2/ 2 )( 1 lim )( 1 lim00 T TT T TT dttg T P dttg T P
  • 15. Are these energy or power signals? •
  • 16. Energy & Power Signals examples • Since in (a) amplitude approaches zero • It’s an energy signal: • Since (b) is periodic, it’s a power signal: t        84422)( 0 20 1 22 2/ dtedtdttgE t 3 1 3 2 1 )( 1 )( 1 lim 32 1 1 1 1 1 21 1 22/ 2/ 2     t dttdttg T dttg T P T TT
  • 17. Worked example • Determine the power of the following signals )cos()(   tCtg             2/ 2/ 22 2/ 2/ 22 2/ 2/ 22/ 2/ 2 2 )]22cos(1[ 2 )(cos 1 )cos( 1 )( 1 lim T T T T T T T TT C dtt T C P dttC T dttC T dttg T P  
  • 18. Work these out • (b) • (c) )cos()cos()( 222111   tCtCtg 21   tj Detg 0 )(  
  • 19. More examples • Compute the signal energy & signal power for the following complex valued signal and indicate whether the signal is an energy or power signal tj Aetg 2 )( 
  • 20. More examples • Since g(t) is a periodic signal, it cannot be an energy signal. Therefore compute the signal power first. Signal Period: • Since the signal has a finite power, it is a power signal & has infinite energy (VERIFY!)  1 0 T   2/122 2 2 1 1 /11 1 /11 1 AtAdtAdtAeP t t tj t t t t       
  • 21. Deterministic & Random Signals • Deterministic signal: Physical description is known in either a mathematical or graphical form. eg: • Random signal: Signal known only in terms of probabilistic description such as mean value, mean square value rather than its complete mathematical or graphical description • eg: noise signal, message signal )1(tan)( 1   ttg
  • 22. Deterministic & Random Signals • Use matlab to plot a deterministic signal: • & random noise • To use matlab you must: • (a) declare the variables • (b) since it’s first time, use the plot command • (c) label the plot )1(tan)( 1   ttg
  • 23. Signal Operations • Time shifting: If a signal g(t) is time shifted by t1 units, it is denoted as f(t) =g(t-t1). • If t1>0, the shift is to the right (time delay) • If t1<0, the shift is to the left (time advance) • To demonstrate time shifting plot the signals: )1(tan)( )1(tan)( )(tan)( 1 1 1       ttY ttG ttg
  • 24. Time Shift matlab code • close all; % close graghs • clear all; % clear all the variables & functions from memory • t = -5:.3:5; % declear variable "t" • g = atan(t); • G = atan(t-1); • Y = atan(t+1); • plot(t,g); % plots the function "g" • hold; • plot(t,G,'r'); % plots “G” • plot(t,Y,'k'); • hold off; • grid on; • xlabel('t'); • ylabel('g(t),G(t) & Y(t)'); • title('time shifting demonstration'); • legend('g-original','G-delay','Y-advance');
  • 26. Signal Operations • Time Scaling: Compression or expansion of a signal • Signal f(t) is g(t) compressed by a factor of ‘a’ if f(t) = g(at), therefore f(t/a) = g(t) for a>1 • Similarly f(t) is g(t) expanded (slowed down) by a factor of ‘a’ if f(t) = g(t/a), therefore f(at) = g(t) for a<1 • To time-scale a signal by a factor of ‘a’, replace t with at. • If a > 1 the scaling is compressed & if a < 1, the scaling is expanded.
  • 27. Time Scaling demonstration • close all; • clear all; • t = -5:.3:5; • f = sawtooth(t); • G = sawtooth(2*t); • F = sawtooth(t/2); • plot(t,f); hold; • plot(t,G,'r'); • grid on; • xlabel('t'); • ylabel('f(t) & G(t)'); • title('time scaling demonstration'); • legend('f','G');
  • 29. Signal Operations • Time Reversal/Inversion/Folding: • To time reverse a signal, replace t with –t • If f(t) is a time resersal of g(t) then • f(t)=g(-t) • See the matlab code of g(-t)
  • 30. Time Reversal Demo-code • close all; clear all; • t = -5:.3:5; • g = atan(t); • G = atan(-t); • plot(t,g); % hold; • plot(t,G,'r'); % hold off; • grid on; • xlabel('t'); • ylabel('g(t) & G(t)'); • title('time Reversal Demonstration'); • legend('f-original', 'G-timeReversed');
  • 32. • Continous time signal: g(t) • Samples of continuous-time signal: g(nT) • Discrete-time signal: g(n)
  • 33. Samples of Continuous-time signal • close all; clear all; • t=-5:.5:5; • g=atan(t); • plot(t,g); • ylabel('g(t)'); • grid on; • title('Continuous-time signal Demonstration'); • figure • stem(t,g); • grid on; • xlabel('nT'); • ylabel('g(nT)'); • title('Samples of Continuous-time signal');
  • 35. Delta Function • Delta/Dirac/Unit Impulse function: Rectangular pulse with an infinitesimally small width & infinitely large height & an overall area of unity.      1)( 0)( dtt t   0t
  • 36. Multiplying a function by delta • Sampling/Sifting property of delta   )()(() )(*)0()()( TtTfTttf tfttf     )()()()()( )0()()0()()( TfdtTtTfdtTttf fdttfdtttf                
  • 37. Sampling/sifting property • The area under the product of a function with delta is equal to the value of that function at the instant where delta is located • Function f(t) must be continuous where the delta is located
  • 38. Unit Step Function       t d 1 0 )(  0 0   t t   t tud )()(  )(t dt du 
  • 39. Time Shifted, scaled, reversed step • Causal function: It’s zero before t = 0 otherwise is non-causal          )( )( )( a b tu a b tu batu 0 0   t t
  • 40. Correlation • Correlation coefficient cn: • Cross-correlation: • Autocorrelation: 11  nc     dttztg EE c zg n )()( 1     dttztggz )()()(      dttgtgg )()()( 
  • 41. Fourier Series • Fourier analysis considers signals to be constructed from a sum of complex exponentials with appropriate frequencies, amplitude & phases • Frequency components are the complex exponentials (sines & cosines) which, when added together, make up the signal • Orthogonality of signal set: ntxtxtx )(),...(),( 21
  • 42. Generalized Fourier Series nm nm        1 2211 )()( ...)(...)()()( n nn nn txctg txctxctxctg 21 ttt  dtxtg Edtx dtxtg c t t n n t t n t t n n       2 1 2 1 2 1 )( 1)( 2      n t t nm E dtxtx 0 )( 2 1 Fourier coefficients nm xx , Are othorgonal
  • 43. Exponential FS • Orthogonality: • Expon. FS       0 )( 0 )( 0 00 0 0 T dtedtee T tnmjtjn T tjm  nm nm          0 0 0 )( 1 )( 0 T tjn n n tjn n dtetg T D eDtg  
  • 44. Parseval’s Theorem • Energy of the sum of orthogonal signals is equal to the sum of their energies: • Parseval’s theorem: 1 2 1 2 1 2 1 2 1 2 1 222 111 2 1 2 1 )()( )....()( );()( EcdttxcdttxcE txctg txctg t t t t    1g n n ng g EcE EcEcE   2 2 2 21 2 1 ...
  • 45. Trigonometric Fourier Series • Trig. FS                  01 1 01 1 01 1 01 1 01 1 0 0 0 0 0 0 0 2 0 0 1 00 2sin)( 2 )( 1 2cos)( 2 2cos 2cos)( 2sin2cosag(t) Tt t n Tt t Tt tTt t Tt t n n n n tdtfntg T b dttg T a tdtfntg Ttdtfn tdtfntg a tfnbtfna     
  • 46. Fourier Transform & Spectra of Aperiodic Signals • The spectrum of a periodic signal is found from FS of a signal over one period • Since the FS is a periodic function of time, it is equal to an aperiodic signal only over the FS expansion interval, outside this interval it repeats • FS is used to produce the spectrum of the periodic extension but not the spectrum of aperiodic signal • To find the spectrum of an aperiodic signal we use FT
  • 47. Fourier Transform (FT) • To develop the FT, let’s start with the exponential FS representation of a periodic signal over the interval -T/2<t<T/2 • Let the interval increase until the entire time axis is encompassed • Since FT is developed from the FS the conditions for the existence follow from those of the Dirichlet conditions    dttg )(
  • 50. Fourier Transform Theorems • FT characteristics are expressed in the form of theorems • The theorems are useful in computing FT of complicated signals • Linearity: • If x(t) X(f) & y(t) Y(f) • Then • ax(t)+by(t) aX(f)+bY(f) • Integral in a linear operation   
  • 51. Fourier Transform Theorems • Time Shift • If Then Prove: 02 0 )()( )()( ftj efXttx fXtx    )(22)(2 0 )(22)(2 0 00 2 00 00 00 )(])([)]([ )()()]([ )()]([ tfjfjtfj tfjfjtfj ftj efXdexettx deexdexttx ttddttt dtettxttx                          
  • 52. Fourier Transform Theorems • Time Reversal )()( )()( fXtx fXtx  
  • 53. Fourier Transform Theorems • Time scale (Scale change) )/( 1 )( )()( afX a atx fXtx  
  • 54. Frequency Translation • If • Then • Modulation • If • then )()( )()( 0 2 0 ffXetx fXtx tfj    )( 2 1 )( 2 1 2cos)( )()( 000 ffXffXtftx fXtx   
  • 55. Inverse FT • Find the inverse FT of the delta function )(2 )(2 2 1 )( 2 1 )( 2 1 )]([ )]([)()( 0 0 0 00 1 12 0 0 0 0                            tj tj tj tjtj ftj e e e ede fGdfefGtg
  • 56. FT of cosine • Find the FT of the sinusoid (cosine) )()(][cos )](2)(2[ 2 1 )]( 2 1 [][cos )( 2 1 cos 000 000 0 00 00           t eet eet tjtj tjtj
  • 57. FT of rectangle • Find the FT of a rectangle of width tau                       /22/0)2/(sin )2/(sin)( )2/(sin)2/sin( 2/ )2/sin( 2 )( 2 )(2 )( 1 )( )()( 2/2/ 2/2/ 2/ 2/ nnc c t rect cX j ee ee j X dtedte t rectX jj jj tjtj              