Signal processing A. Sesieri Diparimeno di Meccanica e Aeronauica Universiy La Sapienza, Rome Presenaion layou - Fourier series and Fourier ransforms - Leakage - Aliasing - Analog versus digial signals - Digial Fourier ransform - FFT - Concluding remarks
FOURIER ANALYSIS DMA Universià di Roma La Sapienza To pass from he ime o he frequency domain, a Fourier ransformaion is necessary: Fourier series for periodic signals Fourier ransform for ransien signals Power specral densiy for random signals d e f c e c f n b n a a f T jn n jn n n n n = = = + + = 0 0 0 1 0 0 0 ) ( ) ( ; } sin cos { ) ( ω ω ω ω ω ω ω ω ω d e F f d e f F j j + + = = ) ( ) ( ) ( ) ( T F F S T T T ff ) ( ) ( lim ) ( * ω ω ω =
FOURIER ANALYSIS DMA Universià di Roma La Sapienza
SOME IMPORTANT PROPERTIES OF F.TRANSFORMS If he ime signal is he convoluion produc of wo funcions, is Fourier ransform is jus he produc of he Fourier ransforms of he wo signals If he ime signal is he produc wo funcions, is Fourier ransform is jus he convoluion produc of he wo Fourier ransforms Y ( f ) =
SOME IMPORTANT PROPERTIES OF F.TRANSFORMS 0 0 f 0 0 ± jω ± jω 1 e d = δ ( ω ) δ ( ) e d = 1 Thus, frequency and ime are perfec dual spaces f
FOURIER ANALYSIS (con d) DMA Universià di Roma La Sapienza The convergence of Fourier series and Fourier ransforms are assured by he Dirichle condiions. However, since he experimenal daa are always runcaed signals, i.e. ransiens, only he Fourier ransform is used for hem. Moreover, he inroducion of Dirac δ permis o overcome he limiaions provided by he Dirichle condiions. E.g.
POWER AND ENERGY SIGNALS: PARSEVAL S THEOREMS Assume x() is a periodic ension, i.e. The average power dissipaed across a uni resisance is: Assume x() is a non periodic ension, i.e. The oal energy dissipaed across a uni resisance is Thus, periodic signals are called power signals while non periodic signals are called energy signals
ERROR DUE TO DATA TRUNCATION: LEAKAGE Experimenal daa are known only for a shor period of ime, e.g. beween -T/2 and T/2. We can imagine hey are seen hrough a window w() such ha w()
LEAKAGE con d The runcaed signal is x T () = x() w(), whose FT is The window leads o a bias error and he convoluion inegral implies a disorsion on X(g), ha spreads over he whole frequency range. Such disorsion is called leakage and, due o he sidelobes of W(f), he X(g) frequency componens spread over he frequency range
LEAKAGE (con d) DMA Universià di Roma La Sapienza
ALIASING DMA Universià di Roma La Sapienza Because of he perfec dualiy beween ime and frequency domains, i is possible o depic he following relaionships.
ALIASING (con d) Therefore he FT of any sampled signal is periodic wih period 1/Δ, i.e. X s (f) = X(f + n/δ) Assuming X(f) = 0 for f Δ/2 Specrum of he real signal X(f) Specrum of he sampled signal 1/Δ 1/2Δ -f o f o 1/2Δ 1/Δ Thus X s (f) = X(f) only for f < Δ/2
ALIASING (con d) However, if X(f) 0 for f Δ/2 Specrum of he aliased signal Thus, o avoid aliasing, one mus choose 1/2Δ > f 0 (maximum frequency of he signal)
ALIASING (con d) Paricularly, by calling - 1/Δ = f s he sampling frequency - 1/2Δ = f f he folding or Nyquis frequency - f 0 he maximun frequency of he signal - T s = 1/f s To avoid aliasing one mus fix f s > 2 f 0 and more precisely fix he acquisiion parameers as follows τ = acquisiion period Δf = frequency resoluion N = number of samples
ALIASING (con d) Aliasing can be also observed in he ime domain If he sampling period is larger han he half of he signal period, T s > T/2, aliasing is observed
ALIASING : example A harmonic signal (100 Hz) sampled wih a sampling frequency of 120 Hz
ANALOG VERSUS DIGITAL SIGNALS DMA Universià di Roma La Sapienza Experimenal daa, deermined by common rasducers, are generally analog signals. For daa processing such daa are always ransformed ino digial daa by an analog o digial converer (A/D) (sampling) x(nδ) n = 0,1, N - 1
ANALOG TO DIGITAL CONVERTER DMA Universià di Roma La Sapienza Available levels = 2 n where n = number of bis Depending on he number of bis of he A/D converer, one always have a a larger or smaller sampling error
EFFECT OF SAMPLING ON LEAKAGE DMA Universià di Roma La Sapienza Acually, because he signal is sampled boh in he ime and frequency domain, we never see he sidelobes ypical of he leakage error, bu raher a se of sampled lines.
LEAKAGE (con d) I can be shown also ha, if he acquisiion ime corresponds exacly o he period of he periodic signal, no leakage is observed (he sampled lines exacly correspond o he zeros of he sidelobes)
LEAKAGE (con d) If T is he period Δf = 1/T
LEAKAGE (con d) Δf = 0.1
LEAKAGE (con d) Δf = 0.1
DIGITAL FOURIER TRANSFORM DMA Universià di Roma La Sapienza For sampled signals i is no possible o carry ou any inegral operaion I is possible however o compue a Discree Fourier Transform ha is defined as: N is oally arbirary. However, for he applicaion of he Fas Fourier Transform (FFT), a very fas algorihm of DFT, N should be chosen appropriaely, i.e. N=2 M where M is an ineger number
CONCLUDING REMARKS In summary here are several errors arising in signal processing. - Leakage is an error due o daa runcaion: i can be avoided or diminished by acquiring as large period of daa as possible, or by using appropriae windows, e.g. Hanning for random daa. - Sampling errors depend on he A/C conversion and can be reduced by increasing number of bis of he A/C converer. - Aliasing is an error due o he sampling mechanism: i can be avoided by choosing appropriaely he acquisiion parameers.
ALIASING (con d) Therefore he FT of any sampled signal is periodic wih period 1/Δ, i.e. X s (f) = X(f + n/δ) Assuming X(f) = 0 for f Δ/2 Specrum of he real signal Specrum of he sampled signal Thus X s (f) = X(f) only for f < Δ/2