Spectra of Digital Waveform - Mosaic Industries1 Spectra of Digital Waveform time-domain and...

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Spectra of Digital Waveformtime-domain and frequency-domain
Prof. Tzong-Lin Wu
Department of Electrical Engineering andGraduate Institute of Communication Engineering
National Taiwan University
Basic Properties of Fourier Transform
0
0
-
0
( )
( )
1. periodic ( )
( ( ) ) 1/ T jn t n
nn
jn t
x t e dt C
x t C
x t
F x t T
e
=
=
=
=

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Basic Properties of Fourier Transform*
- 2. for real ( )n nC C x t=
1 2 1 23. ( ( ) + ( ) ) = ( ( )) + ( ( ))F x t x t F x t F x t

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04. ( ( ) ) ( ( ) )jnF x t F x te =
0n5. ( (t ) ) C (1/ )
jF kT T e = 0, 1, 2, 3....k =

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( )06. ( ( ) / ) ( )
k k k kn nF d x t dt C jn C =

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Spectra of digital circuit waveforms
a. The periodic , trapezoidal pulse train representing clock and data signals of digital systems.
A: Amplitude, f: pulse rise time , : pulse fall timer
: pulse width ( 50 % )
r f
A
A/2
T t
c. To obtain the complex exponential Fourier series of this waveform
Next period
tT
(1)x
x
tT
rA/
r-A/ -A/ f
-A/ frA/
r
r
f
fA/
( ) / 2r f +
( ) / 2r f +

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00
0
0 0 0
( ( ) / 2)
( ( ) / 2)
/ 2 / 2 / 2
( '') / - / - /
/
/ * [ ( / )* ( - )
-
r fr
r f
r r r
jnjnr r f
jnf
jn jn jnr
F x A T A e T A e T
A e T
A T e e e
+
+ +
=
+
=0 00 0
0 0
0
/ 2 / 2/ 2
( ) / 2 / 220 0 0
/ 20 0
( / ) * ( - ) ]
( / 2 )* ( ) * *[sin( / 2)* /( / 2) - sin( / 2)* /( / 2) ]
f fr
r f
jn jnjn jnf
jn jnr r
jnf f
e e e e
j A n n e n e nn e n
+
=
(2)
0
20
( ) / 20 0
0 0
( ) ( '') / ( ) , 0
sin( / 2) sin( / 2) / 2 ( / 2)
r
r f
jnrn
r
F x F x jn nif
n nC A eT n n
+
=
=
=
(3)
d. Expression as one-side spectrum
0 01
0 0
( ) cos( )
when 2
2 sin( / ) /( / ) sin( / ) /( / ) , 0 with , C A /T , and 2 /T
Note : (1) 50% duty cycle
n nn
n n
r r
r f
x t C C n t C
C C
A n T n T n T n T for n
+
=
+
= + +
=
=
= = =
/T =1/2 no even harmonics (2) slight variations from an 50% even harmonics vary widely

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Spectral bounds for trapezoidal waveforms
1 , for small a. sin /
1/ , for small
20 log1/ = 20dB/decade , for small
xx x
x x
x x
sin /x x
20dB/decade
1 2 3x
b. let f = n / T envolope = 2A* sin( f)/( f) sin( f)/( f) / T
20log(envolope) = 20log(2A / T ) + 20log sin( f)/( f) +20log sin( f)/( f)r r
r r
2A /T 20 /db decade
40 /db decade
1/ 1/ rf
Effect of rise/fall time on Spectral Content

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=5 r f ns =
6.37MHz 15.9MHz 63.66MHzf
1v
20ns 5ns
11th harmonic
110MHz
68dBuV 86.1dBuV

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=5 r f ns =
6.37MHz 15.9MHz 63.66MHzf
1v
High frequency parts
. for example : 10MHz , =20 =5 r f r fc ns and ns = =
110MHz 10 1015.9 110Level 20log (1 ) 20log ( ) 40( )6.37 15.9
78.45
MHz MHzVMHz MHz
dB V
=
=
110MHz 10 1063.7 110Level 20 log (1 ) 20 log ( ) 40( )63.7 63.7
90.5
MHz MHzVMHz MHz
dB V
=
=
20ns rise time
5ns rise time
Spectral bound prediction

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The exact value calculated by sinc function
20ns 73.8dBuV 5ns 90.4dBuV
Very close to the spectral bound prediction
The measured values are
20ns 68.8dBuV 5ns 86.1dBuV
Why are they lower than the value by the exact calculations?
What is the effect of the duty cycle on the spectral bound?
Effect of Repetition rate and Duty Cycle
DT
=
What is the effect of the duty cycle on the spectral bound?
0
0
sin( )2 rnr
n fn Dc ADn D n f
+ =
Duty cycle:
One-sided spectrum:
DC level
1st breakpoint

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Effect of Ringing (Undershoot/Overshoot)
0 0 0
0 0
0
/ 2 / 2/ 2
( )0 0
/ 2/ 2
( )0
0 0/ 40
0
1 1sin( ) sin( )
1(1 ) sin( )
1 1sin( ) sin( )4 4
12 24
T Tjn t jn T jn tt t
n n squarewave r r
Tjn T jn tt
n squarewave r
jn T
c c Ke t e dt e Ke t e dtT T
c e Ke t e dtT
n T n TV Ken T
= + + +
= + +
= +
0
0
/ 42 2 2
0
2 2 20/ 40 0
2 2 2
0
1 24
1sin( ) (2 ( / ) ) ( )412 2 ( )4
jn T r
r
jn T r r
r
pep pn T
n TV p K V pe
p pn T
+ + +
+ + + +=
+ + +
0p jn=whereBandpassSquare wave components
Assume = 0
Effect of Ringing (Undershoot/Overshoot)
The spectrum of a square wave with undershoot/overshoot has a part of its spectrum enhanced or increased about the ringing frequency Wr.
Quite often we see in the radiated emission profile a seemingly Resonant region of enhanced emissions in a narrow frequency band.One possible explanation for this is the undershoot/overshoot present on the digital waveforms.

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Use of Spectral Bounds in Computing Bounds on the Output Spectrum of a Linear System
Spectrum Analyzer

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Spectrum Analyzer
Bandwidth of SA
The displayed level at the center frequency of the bandwidth will be the sum of the spectral levels that fall within the bandwidth of the filter at that time.
In order to obtain the lowest possible level on the SA display, we should choose as small a bandwidth as possible.
Spectrum Analyzer
EMC receiver regulation

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Spectrum Analyzer
Spectrum Analyzer
Peak and Quasi-peak
Infrequently occurring signals will result in a measured Quasi-peak level that is considerably smaller than a peak detector would give.
That infrequent events may be of sufficient magnitude distressingly large received levels on a SA that is set to peak detector function.
The reason for the use of quasi-peak detector function relates to the intent of the regulatory limits, which is to prevent interference in radio and wire communicationReceivers. Infrequent spikes and other events do not substantially prevent the listener from obtaining the desired information.

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Representation of Random Signals
01( ) [1 ( )]2
x t X m t= +
Digital data waveforms are obviously random signals, not deterministic signals
/ 2
/ 2
20
20
20
( ) ( ) ( )
1 ( ) ( )
1 [1 ( )][1 ( )]41 [1 ( ) ( ) ( ) ( )]41 [1 ( ) ( )]4
1 for
0 for
limx
T
t T
R x t x t
x t x t dtT
X m t m t
X m t m t m t m t
X m t m t
TT
T
= +
= +
= + + +
= + + + + +
= + +
=
stationary
Representation of Random Signals
Power Spectral density
( ) ( ) jx xG f R e d
=

Spectra of Digital Waveform - Mosaic Industries1 Spectra of Digital Waveform time-domain and...

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