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PHYS3152 Experimental Physics

创建日期：2024-05-25 15:50:18

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标签： PHYS3152

Experimental Physics

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PHYS3152

Experimental Physics
Lecture Note E5
Frequency Analysis of Electronic Signals
2Purpose: understand the basic principles of frequency domain analysis of
electronic signals and study the Fast Fourier Transform (FFT).
Exercise 1: investigate the relationship between the effective sampling
rate and the resulting frequency resolution for the spectral
analysis using the FFT, and the spectral leakage properties
of the rectangular and Hanning windows.
Exercise 2: study aliasing effect
Exercise 3: using the FFT to analyze a square wave and a triangle wave.
Exercise 4: using the FFT to analyze a signal consisting of a sum of two
sinusoids.
Frequency Analysis of Electronic Signals
3How to Study These Signals?
Random signals: Not so random signal:
frequency
dB
(p
ow
er
)
Solution: frequency analysis
frequency
frequency
dB
(V
)
time time
4Take off your glass or watch at
distance: Monroe
Wear glass and watch closely: Einstein
Monroe’s photo: low frequency pass
Einstein’s photo: high spatial frequency
pass
Superposition these two photos.
Albert Einstein or Marilyn Monroe?
5Time Domain and Frequency Domain
 The time-domain representation gives the amplitudes of the signal at
the instants of time during which it was sampled.
 However, in many cases you need to know the frequency content of a
signal rather than the amplitudes of the individual samples.
 The frequency domain graph shows how much of the signal lies within
each given frequency band over a range of frequencies. It can also
include information on the phase shift that must be applied to each
frequency component to recover the original time signal.
(t) V(f)
How to calculate these?
6Integral transform
problem in
transform space
solution in
transform space
solution of
original problem
Original
problem

b
a
dttKtfg ),()()( 
Typical kernels: eit (Fourier) , e-t (Laplace), t-1 (Mellin), ……
Mapping a function f(t) in t-space into another function g() in  space
e.g. t-space: time domain  space: frequency domain
real space Fourier space (e.g. by X-ray scattering)
Some properties of f(t) may be hardly seen in t space, but easily seen in  space
Motivation:
difficult solution
transform reverse transform
easy solution
g() is the integral transform of f(t) by the Kernel K(,t)
7Fourier Transform
Fourier transform
provides the link between
the time-domain and
frequency domain
descriptions of a signal.










dfefVtv
dtetvfV
tfi
tfi
2
2
)()(
)()(


Reverse transform:
8Fourier’s theorem states that any waveform in the
time domain can be represented by the weighted sum
of sines and cosines. The same waveform appears in
the frequency domain as a pair of amplitude and phase
values at each component frequency.
Fourier Series
9One Example
A square wave contains many high-frequency components
10
white noise P(f) = const.
e.g. thermal noise in a circuit…
pink noise P  1/f
earthquake, financial market,
DNA sequence, traffic flow…
Any universal mechanism???
Brown noise P  1/f 2,
Displacement of random walk
(Brownian motion)…
Power Spectrum
V(t) V(f) Power spectrum(a real function)
21( ) ( )lim T
T
P f V f
T

One signal contains many frequency components.
Which components dominate (contribute more energies)?
Fourier
Examples:
11
1 sec continuous wave 1 sec discrete wave
Sampling: the process of converting a signal (for example, a
function of continuous time or space) into a numeric sequence (a
function of discrete time or space).
Sampling of Signal
sampling rate = 256 samples/sec
(i.e. 256 Sa/s, 256 dots per sec)
12
Discrete Fourier Transform (DFT)
t0 : the starting time
 : the smallest sampling time
fs=1/ : sampling rate
T=Nt : sampling duration




1
0
2
0 )()(
N
j
fji
k
kejtvfV  
 The time duration for a signal is finite!
 Assumed periodicity of T=Nt
 The output spectrum takes discrete values: )(
1 ......, ,2 ,1 sk fNN
f 

s
Any difference between FT and DFT?
t0 t0+N
DFT: Fourier transform of discrete functions
13
Fast Fourier Transform (FFT)
 DFT can be defined as where k = 1, 2, …. N
Since both j and k have N values, we need N2 calculations.
 The Fast Fourier Transform (FFT) is simply a fast
(computationally efficient) way to calculate the Discrete Fourier
Transform (DFT). The speed  NlogN.
 In 1969, the 2048 point DFT analysis took 13 ½ hours. Using the
FFT, the same task on the same machine took 2.4 seconds!

 kj
N
iN
j
jk efV
21
0
)(



14
=
Example: FFT of a cosine with 3, 6 or 9 periods
FFT
FFT
FFT
• Broader peaks & Artifacts (finite time effect) • Two peaks (alias)
FFT produces a frequency domain from 0 to fs/2 (from fs/2 to fs is redundant)
     
15
Nyquist Theorem and Aliasing
 Nyquist Theorem: In order to represent the signal unambiguously, it is
necessary to sample at least twice the maximum frequency of the
signal.
 Reversely speaking, if the sampling rate fs is fixed, the highest f can be
correctly sampled is the Nyquist frequency.
 Any analog frequency greater than fs/2 will, after sampling, appear as a
frequency between 0 and fs/2. Such a frequency is known as an “alias”
frequency. So, the frequency fs/2 is called the folding frequency.
2/sNyquist ff 
16
Aliasing
http://www.dsptutor.freeuk.com/aliasing/AliasingDemo.html
The applet is based on a fixed
sampling rate of 8000 samples
per second. The folding
frequency is thus half of
8000 Hz or 4000 Hz.
Case 1) Input frequency fin < fs/2  no aliasing, fout = fin.
Case 2) Input frequency fs/2 < fin < fs  aliasing, with an aliasing
frequency fa = fs – fin..
Case 3) Input frequency f > fs  aliasing, keep subtracting fs from fin
until the result is below fs. Then apply either of the previous
two rules, depending on whether the resulting frequency value
is below or above ½fs.
The following applet demonstrates graphically how an under-sampled
sinusoidal signal appears as though it has a lower "alias" frequency.
e.g. Fix input f, change sampling fs
True f = 8 Hz
Sampling fs = 8 Hz  measured f = 0Hz
Sampling fs = 7 Hz  f = 1Hz,
Sampling fs = 9 Hz  f = 1Hz
fs =10Hz  f = 2Hz,
fs =11Hz  f = 3Hz,
…
fs = 16Hz  f = 8Hz
fs = 17 Hz  f = 8Hz
…
Correct only when fs >2 f
18
Frequency Folding
Folding frequency = fs/2 = 8. k =15, 17, 31, 33, … will fold to k =1
Theoretical understanding of the frequency folding
Recall DFT: and



1
0
2
0 )()(
N
j
jfi
k
kejtvfV   )(1 ......, ,2 ,1 sk fNNf  
)()( * kkN fVfV  / 2 / 2 2
2 NkiNkiiN
kNi
eeee 



 

Fixed fs = 16
(in units of 1/N)
so only frequencies from 0 to fs / 2 are needed.
output f
19
Example: Alias of a Triangular Wave
1MSa/s
500kSa/s
250kSa/s
f =25KHz
20
Ways to Avoid Aliasing
 Ensure that the sampling rate is larger than twice the highest
frequency component present in the signal, if known.
 Ensure that the signal is composed of frequency components
less than fs/2 by using a low-pass filter;
You lose the high frequency components
but ensure that the low frequency
components are properly represented.
21
Finite Time Effect
If not repeat:
*=
FFT
Bad
In practice, we only have finite time records. Such termination leads to the
distortion of the spectrum.
Short signals have broad frequency spectra
Long signals have narrow frequency spectra
duplicate the finite time record over all
time
FFT Good… …
22
Leakage Broadens Spectrum
original signal is the sum of two different sine waves
limitation of measuring time can create broadening of the spectrum
(leakage of frequency!)
23
Leakage
The leakage effect may also lead to aliasing. Since leakage results in a
spreading of the spectrum, the upper frequency may move beyond the
Nyquist frequency, and aliasing may then result.
FFT in the oscilloscope replicates the finite time record over all time
leakage
time
FFT
frequency
24
10 cycles  FFT sharp
good
10.5 cycles  leakage in FFT
bad
An Example of Leakage
25
Using Windows to Avoid Leakage
signal Hanning
window
 =
no leakage, more accurate frequency measurement,
but amplitude accuracy is sacrificed.
Windowed signal,
both ends are 0
…
replicating
26
Other Common “Windows”
or Hanning
Rectangle (when the signal has no leakage)
27
Comparison between Rectangular and Hanning Windows
Fourier Transform of a 1KHz sinusoid.
The 1024 point DFT using the rectangular
window. The large spectral leakage
associated with the rectangular window
makes it a less desirable choice.
The 1024 point DFT using the Hanning
window. By avoiding an abrupt truncation
of the time-domain signal, the spectral
leakage properties are improved. The
main lobe width of the Hanning-windowed
spectrum is wider than that of the
rectangular-windowed spectrum and thus
the spectral resolution of the former is
reduced. The Hanning window is popular
because it achieves a good balance
between spectral leakage and resolution.
(a)
(b)
(c)
dBV=20log(V(f)/1Volt RMS)
28
Effects of Windowing
k = 7, 18
k = 18 not visible
(sampling stops at n = 16)
k = 18 becomes visible
after applying the
Hanning window
Windowing 
• Makes spectrum visible
• Broadens the spectrum
• Modifies the amplitudes of frequency components

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EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 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