FOURIER ANALYSIS PART 1: Fourier Series DISP-2003 FOURIER ANALYSIS PART 1: Fourier Series Maria Elena Angoletta, AB/BDI DISP 2003, 20 February 2003 Introduction to Digital Signal Processing

TOPICS 2. A tour of Fourier Transforms DISP-2003 TOPICS 1. Frequency analysis: a powerful tool 2. A tour of Fourier Transforms 3. Continuous Fourier Series (FS) 4. Discrete Fourier Series (DFS) 5. Example: DFS by DDCs & DSP M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 2 / 24 Introduction to Digital Signal Processing

Frequency analysis: why? Fast & efficient insight on signal’s building blocks. Simplifies original problem - ex.: solving Part. Diff. Eqns. (PDE). Powerful & complementary to time domain analysis techniques. Several transforms in DSPing: Fourier, Laplace, z, etc. time, t frequency, f F s(t) S(f) = F[s(t)] analysis synthesis s(t), S(f) : Transform Pair General Transform as problem-solving tool M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 3 / 24

Fourier analysis - applications Applications wide ranging and ever present in modern life Telecomms - GSM/cellular phones, Electronics/IT - most DSP-based applications, Entertainment - music, audio, multimedia, Accelerator control (tune measurement for beam steering/control), Imaging, image processing, Industry/research - X-ray spectrometry, chemical analysis (FT spectrometry), PDE solution, radar design, Medical - (PET scanner, CAT scans & MRI interpretation for sleep disorder & heart malfunction diagnosis, Speech analysis (voice activated “devices”, biometry, …). M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 4 / 24

Fourier analysis - tools Input Time Signal Frequency spectrum Periodic (period T) Discrete Continuous FT Aperiodic FS Note: j =-1,  = 2/T, s[n]=s(tn), N = No. of samples Discrete DFS Periodic (period T) Continuous DTFT Aperiodic DFT ** Calculated via FFT M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 5 / 24

A little history Astronomic predictions by Babylonians/Egyptians likely via trigonometric sums. 1669: Newton stumbles upon light spectra (specter = ghost) but fails to recognise “frequency” concept (corpuscular theory of light, & no waves). 18th century: two outstanding problems celestial bodies orbits: Lagrange, Euler & Clairaut approximate observation data with linear combination of periodic functions; Clairaut,1754(!) first DFT formula. vibrating strings: Euler describes vibrating string motion by sinusoids (wave equation). BUT peers’ consensus is that sum of sinusoids only represents smooth curves. Big blow to utility of such sums for all but Fourier ... 1807: Fourier presents his work on heat conduction  Fourier analysis born. Diffusion equation  series (infinite) of sines & cosines. Strong criticism by peers blocks publication. Work published, 1822 (“Theorie Analytique de la chaleur”). M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 6 / 24

A little history -2 19th / 20th century: two paths for Fourier analysis - Continuous & Discrete. CONTINUOUS Fourier extends the analysis to arbitrary function (Fourier Transform). Dirichlet, Poisson, Riemann, Lebesgue address FS convergence. Other FT variants born from varied needs (ex.: Short Time FT - speech analysis). DISCRETE: Fast calculation methods (FFT) 1805 - Gauss, first usage of FFT (manuscript in Latin went unnoticed!!! Published 1866). 1965 - IBM’s Cooley & Tukey “rediscover” FFT algorithm (“An algorithm for the machine calculation of complex Fourier series”). Other DFT variants for different applications (ex.: Warped DFT - filter design & signal compression). FFT algorithm refined & modified for most computer platforms. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 7 / 24

For all t but discontinuities Fourier Series (FS) * see next slide A periodic function s(t) satisfying Dirichlet’s conditions * can be expressed as a Fourier series, with harmonically related sine/cosine terms. a0, ak, bk : Fourier coefficients. k: harmonic number, T: period,  = 2/T For all t but discontinuities synthesis analysis (signal average over a period, i.e. DC term & zero-frequency component.) Note: {cos(kωt), sin(kωt) }k form orthogonal base of function space. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 8 / 24

if s(t) discontinuous then |ak|<M/k for large k (M>0) FS convergence s(t) piecewise-continuous; s(t) piecewise-monotonic; s(t) absolutely integrable , (a) (b) (c) Dirichlet conditions In any period: if s(t) discontinuous then |ak|<M/k for large k (M>0) Rate of convergence Example: square wave (a) (b) (c) M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 9 / 24

FS of odd* function: square wave. FS analysis - 1 FS of odd* function: square wave. (zero average) (odd function) * Even & Odd functions Odd : s(-x) = -s(x) Even : s(-x) = s(x) M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 10 / 24

FS analysis - 2 Fourier spectrum representations fk=k /2 rK = amplitude, K = phase vk = rk cos (k t + k) Polar Rectangular vk = akcos(k t) - bksin(k t) Fourier spectrum of square-wave. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 11 / 24

Square wave reconstruction from spectral terms FS synthesis Square wave reconstruction from spectral terms Convergence may be slow (~1/k) - ideally need infinite terms. Practically, series truncated when remainder below computer tolerance ( error). BUT … Gibbs’ Phenomenon. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 12 / 24

Overshoot exist @ each discontinuity Gibbs phenomenon Overshoot exist @ each discontinuity Max overshoot pk-to-pk = 8.95% of discontinuity magnitude. Just a minor annoyance. FS converges to (-1+1)/2 = 0 @ discontinuities, in this case. First observed by Michelson, 1898. Explained by Gibbs. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 13 / 24

/2-advanced square-wave FS time shifting FS of even function: /2-advanced square-wave (even function) (zero average) phase amplitude Note: amplitudes unchanged BUT phases advance by k/2. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 14 / 24

Complex FS Euler’s notation: e-jt = (ejt)* = cos(t) - j·sin(t) “phasor” analysis Complex form of FS (Laplace 1782). Harmonics ck separated by f = 1/T on frequency plot. synthesis Note: c-k = (ck)* Link to FS real coeffs. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 15 / 24

* Explained in next week’s lecture FS properties Time Frequency * Explained in next week’s lecture Homogeneity a·s(t) a·S(k) Additivity s(t) + u(t) S(k)+U(k) Linearity a·s(t) + b·u(t) a·S(k)+b·U(k) Time reversal s(-t) S(-k) Multiplication * s(t)·u(t) Convolution * S(k)·U(k) Time shifting Frequency shifting S(k - m) M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 16 / 24

FS - “oddities” Orthonormal base Fourier components {uk} form orthonormal base of signal space: uk = (1/T) exp(jkωt) (|k| = 0,1 2, …+) Def.: Internal product : uk  um = δk,m (1 if k = m, 0 otherwise). (Remember (ejt)* = e-jt ) Then ck = (1/T) s(t)  uk i.e. (1/T) times projection of signal s(t) on component uk Orthonormal base k = - , … -2,-1,0,1,2, …+ , ωk = kω, k = ωkt, phasor turns anti-clockwise. Negative k  phasor turns clockwise (negative phase k ), equivalent to negative time t,  time reversal. Negative frequencies & time reversal Careful: phases important when combining several signals! M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 17 / 24

FS - power Average power W : Example Parseval’s Theorem FS convergence ~1/k  lower frequency terms Wk = |ck|2 carry most power. Wk vs. ωk: Power density spectrum. Example Wk = 2 W0 sync2(k ) W0 = ( sMAX)2 sync(u) = sin( u)/( u) Pulse train, duty cycle  = 2 t / T bk = 0 a0 =  sMAX ak = 2sMAX sync(k ) M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 18 / 24

FS of main waveforms M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 19 / 24

Discrete Fourier Series (DFS) Band-limited signal s[n], period = N. DFS generate periodic ck with same signal period Note: ck+N = ck  same period N i.e. time periodicity propagates to frequencies! DFS defined as: ~ Kronecker’s delta Orthogonality in DFS: analysis Synthesis: finite sum  band-limited s[n] synthesis N consecutive samples of s[n] completely describe s in time or frequency domains. M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 20 / 24

DFS of periodic discrete DFS analysis s[n]: period N, duty factor L/N DFS of periodic discrete 1-Volt square-wave amplitude phase Discrete signals  periodic frequency spectra. Compare to continuous rectangular function (slide # 10, “FS analysis - 1”) M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 21 / 24

* Explained in next week’s lecture DFS properties Time Frequency Homogeneity a·s[n] a·S(k) Additivity s[n] + u[n] S(k)+U(k) Linearity a·s[n] + b·u[n] a·S(k)+b·U(k) Multiplication * s[n] ·u[n] Convolution * S(k)·U(k) Time shifting s[n - m] Frequency shifting S(k - h) * Explained in next week’s lecture M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 22 / 24

DFS analysis: DDC + ... (1) (2) (3) s(t) periodic with period TREV (ex: particle bunch in “racetrack” accelerator) tn = n/fS , n = 1, 2 .. NS , NS = No. samples (1) (2) I[tn ]+j Q[tn ] = s[tn ] e -jLOtn (3) I[tp ]+j Q[tp ] p = 1, 2 .. NT , Ns / NT = decimation. (Down-converted to baseband). M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 23 / 24

Example: Real-life DDC ... + DSP Fourier coefficients a k*, b k* harmonic k* = LO/REV DDCs with different fLO yield more DFS components Example: Real-life DDC M. E. Angoletta - DISP2003 - Fourier analysis - Part 1: Fourier Series 24 / 24