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- 1. Spectral Analysis &The Fourier Transform Athanasios AnastasiouSignal Processing and MultimediaCommunications Research Group University of Plymouth - UK
- 2. Learning Objectives• Familiarise yourselves with the Fourier Transform and its properties• Make sense of Fourier spectra• Carry out basic operations on various signals in MATLAB – Forward / Inverse Transform – Visualisation of spectra
- 3. Topics• Signals In: – Time Domain – Frequency Domain• The Fourier Transform• The Fourier Spectra• Practical application in MATLAB
- 4. Fourier Transform (?)• It was named after Jean Baptiste Joseph Fourier (1768 – 1830)• The Fourier Transform – Changes the representation of a signal – Fundamental characteristics of the signal (power, phase) remain the same, only their representation changes – Representation (?) • Time Domain • Frequency Domain – Enables us to observe signals and change them far more easily in the Frequency Domain, than it would be in the Time Domain.• The same concepts apply similarly to other transforms as well
- 5. What is so special about the Fourier Transform?• An extremely versatile tool• Time Series Analysis / Processing / Synthesis – Tidal observations – Mechanical Vibration Analysis – Lossy Compression – Image Reconstruction – Signal Features / Classification • Speech Recognition / Animal Species Recognition – Electromagnetic Spectrum • SETI• …and some special cases: – Fast Fluid Simulation (for computer graphics)
- 6. Time Domain Frequency Domain• One Signal – x(n) – Amplitude samples of some quantity over time• Many Different Representations – X(ω) – A series of weights to: • Trigonometric functions • Other functions• How do we pass from one domain to the other?
- 7. Integral Transforms Transformed Signal Signal Kernel (Time Domain)(Some Domain) t2 X U t, x(t ) dt Forward Transform t1 t2 1 xt U ,t X d Inverse Transform t1 What about the Fourier Transform?
- 8. The Fourier TransformFrequency Domain Kernel Time Domain i2 t X e x t dt ei cos i sin Euler’s Formula Re-expresses the signal x(t) as a series of weights X(ω) of sinusoids and co-sinusoids of different frequencies ω
- 9. The Inverse Fourier TransformTime Domain Kernel Frequency Domain xt ei 2 t X d Re-composes the signal x(t) as a remix of sinusoids and co-sinusoids at different frequencies (ω) and different “strengths” X(ω)
- 10. The Discrete Fourier Transform Remember, DSP is all about sampled signals… N 1 i2 k n 1 N k e xn N n 0 k N 1 i2 k n N xn e X k k 0…and this has some consequences on the properties of the Fourier Transform (Key properties: Linearity, Periodicity, Shifting, Convolution Duality, Interpolation, Frequency & Time Scaling) Please Note:This is the 1-Dimensional case of the Fourier Transform. It can be generalised to higher dimensions with applications to Image Processing, Volumetric Data processing and others
- 11. Digression: Fast Fourier Transform The Discrete Fourier Transform (DFT) as a matrix operation X U x Re U Im U Perform The Unique Operations Only
- 12. Fourier Spectra• “Spectrum” is another name for X(k) – The strength X(k) by which sinusoids and co-sinusoids of frequency k contribute to signal x(t)• “Fourier Spectrum” is synonymous with “Signal Spectrum” – Usually, “Spectral Analysis” refers to observing the Fourier Spectra of a signal• Hold on….Fourier Spectra?...There’s more than one? – Oh, yeah!
- 13. Fourier Spectra Ak X k Amplitude 2 Pk X k Power 1 Im X k Phase k tan Re X kPlease note:• X(k) is a complex number• The power spectrum is not the same as the Amplitude spectrum!• How do these “spectra” look like and how do we make sense of them?
- 14. Fourier Spectra Amplitude Spectrum Time Domain Frequency Domain 5 1 10 0.8 0.6 0 10 0.4 0.2 X(k)x(t) -5 0 10 -0.2 -0.4 -10 10 -0.6 -0.8 -15 -1 10 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 0.5 1 1.5 2 2.5 time (sec) Frequency x 10 4 What are the units of the Frequency Axis? Where are the 0, Fs/2 and Fs points on the Frequency Axis? What can the units of X(k) be?
- 15. Fourier Spectra Amplitude Spectrum Time Domain Frequency Domain 0 1 10 0.8 -2 10 0.6 -4 10 0.4 -6 10 0.2 -8 10 X(k)x(t) 0 -10 10 -0.2 -12 10 -0.4 -14 10 -0.6 -16 -0.8 10 -18 -1 10 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 1000 2000 3000 4000 5000 6000 7000 8000 time (sec) Frequency (Hz) Physical Frequency k VS f Fs Normalised Frequency N Any general observations on X(k)? What about the relationship between X(k) and x(t)?
- 16. Fourier Spectra Spectral Resolution Time Domain Frequency Domain 0 1 10 0.8 -2 10 0.6 -4 10 0.4 -6 10 0.2 -8 10 X(k)x(t) 0 -10 10 -0.2 -12 10 -0.4 -14 10 -0.6 -16 -0.8 10 -18 -1 10 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 1000 2000 3000 4000 5000 6000 7000 8000 time (sec) Frequency (Hz) Physical Frequency k VS f Fs Normalised Frequency N What happens when some (f) falls between two kn,kn+1?
- 17. Fourier Spectra Digression: Windowing Functions To reduce spectral leakage in x(n): x* n xn W n X* k F x n *F W n Where W(n) is some “Window Function” 2 5 1 10 1 10 0.8 0 0.8 10 0 10 0.6 0.6 -2 A A 10A A 0.4 0.4 -5 10 -4 10 0.2 0.2 -10 -6 0 10 0 10 0 50 100 150 0 0.1 0.2 0.3 0.4 0.5 0 50 100 0 0.1 0.2 0.3 0.4 0.5 n k n k Hamming Window Blackman Window
- 18. Fourier Spectra Power Spectrum Time Domain Frequency Domain 0 1 10 0.8 -5 10 0.6 -10 0.4 10 0.2 -15 10 P(k)x(t) 0 -20 10 -0.2 -0.4 10 -25 -0.6 -30 10 -0.8 -35 -1 10 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 1000 2000 3000 4000 5000 6000 7000 8000 time (sec) Frequency (Hz) What does P(k) shows and what could be its units?
- 19. Fourier SpectraPower Spectrum (Parseval’s Theorem) Time Domain Frequency Domain 0 1 10 0.8 -5 10 0.6 -10 0.4 10 0.2 -15 10 P(k)x(t) 0 -20 10 -0.2 -0.4 10 -25 -0.6 -30 10 -0.8 -35 -1 10 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 1000 2000 3000 4000 5000 6000 7000 8000 time (sec) Frequency (Hz) N 1 N 1 2 1 2 xn X k n 0 N k 0
- 20. Fourier Spectra (Phase) Time Domain Frequency Domain 1 4 0.8 3 0.6 2 0.4 0.2 1x(t) phi(k) 0 0 -0.2 -1 -0.4 -2 -0.6 -0.8 -3 -1 -4 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 1000 2000 3000 4000 5000 6000 7000 8000 time (sec) Frequency (Hz) What does the φ(k) axis shows?
- 21. Fourier Spectra (Phase) Time Domain Frequency Domain 4 1 3 2 1 0.8 phi(k) 0 3 -1 -2 0.6 -3 -4 0 1000 2000 3000 4000 5000 6000 7000 8000 Frequency (Hz) 0.4 2 0.2x(t) 0 1 -0.2 phi(k) -0.4 0 -0.6 -0.8 -1 X: 440 Y: -1.571 -1 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 time (sec) -2 -3 380 400 420 440 460 480 500 520 540 Frequency (Hz) What is the phase of x(n)?
- 22. Fourier Spectra Summary Harmonics Fundamental 5 10 0 10 |A(k)| -5 10 1 -10 10 0.5 -15 10 0 2000 4000 6000 8000x(n) 0 Frequency (Hz) X(k)=F(x(n)) 4 3 -0.5 2 1 -1 phi(k) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 time (sec) x 10 -4 -1 -2 -3 -4 0 2000 4000 6000 8000 Frequency (Hz) 0 Hz Fs/2 Hz FsHz Time Domain Frequency Domain
- 23. Digression (1): Phase & Phase Shift• Positive Phase Shift – x(n)=x(n+a) – Phase Is Advancing – Signal Appears Shifted To The LEFT!• Negative Phase Shift – x(n)=x(n-a) – Phase Is Lagging – Signal Appears Shifted To The RIGHT!• This is also important for “Filter Causality”
- 24. Digression(2): The Phase Spectrum!• The Phase Spectrum is extremely important – It encodes the “structure” of the signal – Two signals can have the same A(k)=A1(k)=A2(k) but different φ1(k), φ2(k) • This implies two completely different x1(n), x2(n)• Practical Example – Draw bars on an Organ – Additive Sound Synthesis – The importance of phase in speech enhancement
- 25. Fourier Transform• Practical Application
- 26. Spectral Analysis & The Fourier Transform• We will be using MATLAB• Create / Acquire Some Signals• Perform Forward / Inverse Fourier Transform• Visualise and read the spectra of some basic signals• Explore some properties of the Fourier Transform – And their consequences to DSP• Brief Case Study On Sound Analysis
- 27. Creating / Acquiring Some Signals• Task 1 - For some… – Sampling Fequency Fs (Hz) – Signal Duration D (Sec) – Fs=8000; D=3; n=0:(1./Fs):D; p=2*pi*n; • What are the (n) and (p)?• …generate some basic signals (for some frequency f of your choice) – x1=sin(f*p); #Sin – x2=cos(f*p); #Cos – x3=square(f*p) #... – x4=sawtooth(f*p); #... – x5=sawtooth(f*p,0.5); #Triangle – x6=zeros(size(t));x6(1)=1; #Delta
- 28. Creating / Acquiring Some Signals• Task 1.1 - Other signals? – Text Format • MATLAB can load a list of numbers very easily through the command “load” • Start “Notepad”, enter a coma separated list of numbers, save the file in your current path, load the file in MATLAB. – WAV Format • Sound files can be acquired from “freesound.org” and be loaded into your workspace through “wavread” • Locate a .wav file, load it into matlab using [x7, Fs]=wavread(‘path/to/some/file.wav’);
- 29. Digression: Visualising / Sonifying Some Signals• Visualisation – Any time series data • plot(x*); – plot(n,x*) • stem(x*);• Sonification – How does a time series of values sounds like? – sound(x*,Fs) – soundsc(x*,Fs)
- 30. The Forward / Inverse Fourier Transform• Select some signal (x*) from your workspace• Perform Forward Fourier Transform – Fx*=fft(x*);• Perform Inverse Fourier Transform – IFx*=ifft(Fx*)• Please note: The “*” symbol stands for some signal created in the previous step• What can we observe about Fx*?• Was IFx* the same as x?
- 31. Visualising Signal Spectra• Task 2 – Select some Fx* from your workspace – Derive its Amplitude spectrum • A(Fx*) = abs(Fx*); – Derive its Power spectrum • P(Fx*) = abs(Fx*)^2; – Derive its Phase Spectrum • Phi(Fx*) = angle(Fx*);• How do they look like? – How can we improve their presentation?• How does the spectrum of the “cos” function looks like? – How is it different to the sin spectrum?• How does the spectrum of the “Delta” function looks like? – Why? What does this mean?• How does the spectrum of the “square” signal looks like?
- 32. Exploring Some Properties Of The Fourier Transform• Shifting In Time / Frequency – How would you shift your x(n) by 5 samples in the time domain? • x(n)=x(n+5); %Yes…I know… • But how can it be done through the Fourier Transform?• After you have shifted your signal – What was put in the place of these 5 values? – Why?
- 33. Brief Case Study Of Real World Signal Spectra 200 150 100Amplitude 50 0 -50 -100 -150 0 0.5 1 1.5 2 2.5 3 Sample (n) x 10 4 6 10 5 10 4 10 Amplitude 3 10 2 10 1 10 0 10 0 20 40 60 80 100 120 Frequency (Hz) Human EEG: “Reading” The Spectrum Signals courtesy of Javier Escudero Rodriguez
- 34. What Does The Spectrum “Tells” Us 2Power Line Amplitude Hum / 0 Other E.M.I. -2 0 20 40 60 80 100 120 Sample (n) 200 Actual Amplitude Signal 0 From the + xn Brain -200 0 0.5 1 1.5 2 2.5 3 Sample (n) x 10 4 Heart’s Amplitude 10 Electrical Activity 0 -10 7000 7500 8000 8500 9000 Sample (n)
- 35. Brief Case Study Of Real World Signal Spectra 0.6 0.4 0.2 Amplitude 0 -0.2 -0.4 -0.6 -0.8 0 500 1000 1500 2000 2500 3000 3500 Sample (n) 5 10 0 10Amplitude -5 10 -10 10 -15 10 0 10 20 30 40 50 60 70 80 90 100 Frequency (Hz) Pendulum Net Acceleration (on Earth): “Reading” The Spectrum
- 36. What Does The Spectrum “Tells” Us• Data Acquisition Fault – The MCU was not sampling the Accelerometer while writing data to the SD card. – Phase Modulation• Very limited actual bandwidth• Did not have a huge impact on the actual application because of the nature of the signal
- 37. Questions for further experimentation• What is the meaning of the DC component? (0Hz) – As far as sampled signals are concerned what other condition can lead to a sampled signal of 0Hz Frequency?• What happens when the FT is evaluated over less / more frequency points than there are samples in a signal?• How would the spectra change if the FS was increased / decreased?• If some x(n) expresses calibrated “Sea Level” measurements sampled every day in a particular location, what is the unit of the Frequency Axis (k) and what is the interpretation of the spectrum? – Is the Frequency Axis still reading in Hz? If not, what can it be interpreted in?• If some x(n) expressed “Hours Of Sleep” measurements sampled every night for a particular person over the course of 10 years, what is the unit of the Frequency Axis (k) and what is the interpretation of the spectrum?
- 38. The End
- 39. Appendix: Crash Course On Complex Numbers Real Part Re(Z) Imaginary Part Im(Z) The Complex Number (Z): Z a i b You can think of it as a vector i 1 And because Z is a vector, it has “Magnitude” and “Orientation” (angle) b rZ a 2 b2 Z tan a And through these, Z can be expressed in “Polar Form”: z rZ cos Z i sin or z rZ ei ZThe basic operations between two Complex Numbers Z,Y in each form are as follows: Z a i b, Y c i d z rZ e i Z , y rY e i Y Z Y a c i b d z y rZ rY e i Z Y Z Y ac bd i bc ad Z ac bd bc ad z rZ i i e Z Y Y c2 d 2 c2 d 2 y rY
- 40. Appendix:Properties Of The Fourier Transform Please Note: In all of the following properties, capitalised letters are the Fourier Transforms of the corresponding small case time series Periodicit y X k N X kConvolutio n xn yn X k Y k and x n y n X k Y k i2 i2 nm nm N N Shifting X k m xn e and x n m X k e Linearity a xn b yn a X k b Y k 1 k Scaling xa n X a a

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