Analog & digital signals. Analog & digital signals
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1 Analog & digital signals Analog & digital signals Analog Continuous function V of continuous variable t (time, space etc) : V(t). Digital Discrete function Vk of discrete sampling variable tk, with k = integer: Vk = V(tk)..3.3 Voltage [V] time [ms] Voltage [V] Uniform (periodic) sampling. Sampling frequency f S = / t S t s t s sampling time, t k [ms]
2 Digital vs analog proc ing Digital vs analog proc ing Digital Signal Processing (DSPing) Advantages More flexible. Often easier system upgrade. Data easily stored. Better control over accuracy requirements. Reproducibility. Limitations A/D & signal processors speed: wide-band signals still difficult to treat (real-time systems). Finite word-length effect. Obsolescence (analog electronics has it, too!).
3 Digital system example Digital system example General scheme Sometimes steps missing - Filter + A/D (ex: economics); - D/A + filter (ex: digital output wanted). V V A A ms ms k Filter Antialiasing A/D Digital Processing ANALOG DOMAIN DIGIAL DOMAIN V k D/A V ms ms Filter Reconstruction ANALOG DOMAIN
4 Digital system implementation Digital system implementation ANALOG INPU KEY DECISION POINS: Analysis bandwidth, Dynamic range Antialiasing Filter A/D Pass / stop bands. Sampling rate. No. of bits. Parameters. 2 Digital Processing DIGIAL OUPU Digital format. What to use for processing? See slide DSPing aim & tools 3
5 Sampling How fast must we sample * a continuous signal to preserve its info content? Ex: train wheels in a movie. 25 frames (=samples) per second. rain starts wheels go clockwise. rain accelerates wheels go counter-clockwise. Why? Frequency misidentification due to low sampling frequency. * Sampling: independent variable (ex: time) continuous discrete. Quantisation: dependent variable (ex: voltage) continuous discrete. Here we ll talk about uniform sampling.
6 Sampling - 2 Sampling t s(t) = sin(2πf t) f S f = Hz, f S = 3 Hz s (t) = sin(8πf t) s2 (t) = sin(4πf t) f S represents exactly all sine-waves s k (t) defined by: s k (t) = sin( 2π (f + k f S ) t ), k
7 he sampling theorem he sampling theorem heo* A signal s(t) with maximum frequency f MAX can be recovered if sampled at frequency f S > 2 f MAX. * Multiple proposers: Whittaker(s), Nyquist, Shannon, Kotel nikov. Naming gets confusing! Nyquist frequency (rate) f N = 2 f MAX or f MAX or f S,MIN or f S,MIN /2 Example s(t) = 3 cos(5 π t) + sin(3 π t) cos(π t) Condition on f S? F F 2 F 3 F =25 Hz, F 2 = 5 Hz, F 3 = 5 Hz f S > 3 Hz f MAX
8 Frequency domain (hints) Frequency domain (hints) ime & frequency: two complementary signal descriptions. Signals seen as projected onto time or frequency domains. Example Ear + brain act as frequency analyser: audio spectrum split into many narrow bands low-power sounds detected out of loud background. Bandwidth: indicates rate of change of a signal. High bandwidth signal changes fast.
9 Sampling low-pass signals Sampling low-pass signals (a) Continuous spectrum (a) Band-limited signal: frequencies in [-B, B] (f MAX = B). (b) -B B f Discrete spectrum No aliasing -B B f S /2 f (b) ime sampling frequency repetition. f S > 2 B no aliasing. (c) Discrete spectrum Aliasing & corruption (c) f S 2 B aliasing! f S /2 f Aliasing: signal ambiguity in frequency domain
10 Antialiasing filter Antialiasing filter (a) Out of band noise Signal of interest Out of band noise (a),(b) Out-of-band noise can alias into band of interest. Filter it before! (b) (c) Passband frequency -B B f -B B f S /2 f Antialiasing filter -B B f (c) Antialiasing filter Passband: depends on bandwidth of interest. Attenuation A MIN : depends on ADC resolution ( number of bits N). A MIN, db ~ 6.2 N +.76 Out-of-band noise magnitude. Other parameters: ripple, stopband frequency...
11 2 ADC - Number of bits N ADC - Number of bits N Continuous input signal digitized into 2 N levels. 3 Uniform, bipolar transfer function (N=3) V Quantization step q = V FSR 2 N Ex: V FSR = V, N = 2 q = 244. µv V FSR LSB Voltage ( = q) Scale factor (= / 2 N ).5 q / 2 Percentage (= / 2 N ) q / 2 Quantisation error
12 2 ADC - Quantisation error ADC - Quantisation error.3 Voltage [V] time [ms] Quantisation Error e q in [-.5 q, +.5 q]. e q limits ability to resolve small signal. Higher resolution means lower e q.
13 Frequency analysis: why? 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. he brain does it? time, t s(t) analysis F synthesis General ransform as problem-solving tool frequency, f S(f) = F[s(t)] s(t), S(f) : ransform Pair
14 Fourier analysis - tools Fourier analysis - tools 2.5 Input ime Signal Frequency spectrum time, t Continuous Periodic (period ) Aperiodic FS F Discrete Continuous c jk ω t k = s(t) e dt + j2 π f t S(f) = s(t) e dt time, t time, t k time, t k 8 2 Discrete Periodic (period ) Aperiodic DFS DF DF ** ** Discrete Continuous Discrete c ~ k = N N j s[n] e n= N j s[n] e n= 2πk n N + S(f) = s[n] e j2π f n n= 2πk n c ~ k = N N Note: j = -, ω = 2π/, s[n]=s(tn), N = No. of samples ** Calculated via FF
15 A little history A little history Astronomic predictions by Babylonians/Egyptians likely via trigonometric sums. 669: Newton stumbles upon light spectra (specter = ghost) but fails to recognise frequency concept (corpuscular theory of light, & no waves). 8 th century: two outstanding problems celestial bodies orbits: Lagrange, Euler & Clairaut approximate observation data with linear combination of periodic functions; Clairaut,754(!) first DF formula. vibrating strings: Euler describes vibrating string motion by sinusoids (wave equation). 87: 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, 822 ( heorie Analytique de la chaleur ).
16 A little history -2 A little history -2 9 th / 2 th century: two paths for Fourier analysis - Continuous & Discrete. CONINUOUS Fourier extends the analysis to arbitrary function (Fourier ransform). Dirichlet, Poisson, Riemann, Lebesgue address FS convergence. Other F variants born from varied needs (ex.: Short ime F - speech analysis). DISCREE: Fast calculation methods (FF) 85 - Gauss, first usage of FF (manuscript in Latin went unnoticed!!! Published 866) IBM s Cooley & ukey rediscover FF algorithm ( An algorithm for the machine calculation of complex Fourier series ). Other DF variants for different applications (ex.: Warped DF - filter design & signal compression). FF algorithm refined & modified for most computer platforms.
17 Fourier Series (FS) Fourier Series (FS) A periodic function s(t) satisfying Dirichlet s conditions * can be expressed as a Fourier series, with harmonically related sine/cosine terms. synthesis synthesis + s(t) = a + k= analysis analysis a a k -bk = = = 2 2 [ ak cos(k ω t) bk sin(k ω t) ] For all t but discontinuities s(t)dt s(t) cos(k ω t)dt s(t) sin(k ω t)dt a, a k, b k : Fourier coefficients. k: harmonic number, : period, ω = 2π/ (signal average over a period, i.e. DC term & zero-frequency component.) Note: {cos(kωt), sin(kωt) } k form orthogonal base of function space. * see next slide
18 FS convergence FS convergence Dirichlet conditions (a) s(t) piecewise-continuous; In any period: (b) (c) s(t) piecewise-monotonic; s(t) absolutely integrable, s(t) dt < Example: square wave Rate of convergence if s(t) discontinuous then a k <M/k for large k (M>) s(t) (a) (b) (c)
19 FS analysis - FS analysis - FS of odd* function: square wave. π 2π a = dt ( )dt = 2π + π π 2π a k = coskt dt coskt dt = π π = 2π ω = (zero average) (odd function) square signal, sw(t) π t -b k π 2π 2 sinkt dt sinkt dt =... = { cos }= π k π π = kπ * Even & Odd functions s(x) = 4 k π,, k odd k even Even : s(-x) = s(x) s(x) x sw(t) = sin t + sin 3 t + sin 5 t +... π 3 π 5 π Odd : s(-x) = -s(x) x
20 FS synthesis FS synthesis Square wave reconstruction from spectral terms sw7(t) sw3(t) sw9(t) 5(t) = [ [ -bk kk sin(kt) ] ] k = square signal, sw(t) Convergence may be slow (~/k) - ideally need infinite terms. Practically, series truncated when remainder below computer tolerance ( error). BU Gibbs Phenomenon. t
21 Gibbs phenomenon Gibbs phenomenon.5 Overshoot each discontinuity 79 [ -bk ] sw79(t) = sin(kt) k= square signal, sw(t) t First observed by Michelson, 898. Explained by Gibbs. Max overshoot pk-to-pk = 8.95% of discontinuity magnitude. Just a minor annoyance. FS converges to (-+)/2 discontinuities, in this case.
22 FS time shifting FS time shifting FS of even function: π/2-advanced squares quare-wave a = -b k 4 k π ak = 4 k π =, (zero average), k odd, k odd,, k =, 5, 9... k = 3, 7,... k even. (even function) square signal, sw(t) amplitude amplitude r k 4/π 4/3π θ k 2 π t f 3f 5f 7f f Note: amplitudes unchanged BU phases advance by k π/2. phase phase π f 3f 5f 7f f
23 Complex FS Complex FS Euler s notation: e -jt = (e jt )* = cos(t) - j sin(t) ck analysis analysis s(t) = jk ω t ck e k= synthesis synthesis = s(t) e -jk ω t dt phasor cos(t) = e jt jt + e 2 sin(t) = e 2 j Complex form of FS (Laplace 782). Harmonics c k separated by f = / on frequency plot. e jt jt Note: c -k = (c k )* z = r e jθ c = ck = a 2 2 ( + j ) = ( a j b ) ak Link to FS real coeffs. bk k k b r θ a r = a 2 + b 2 θ = arctan(b/a)
24 FS properties FS properties ime Frequency 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) ime reversal s(-t) S(-k) Multiplication * Convolution * ime shifting s(t) u(t) s(t t) u( t)dt s(t t) 2π m t + j m= S(k m)u(m) S(k) U(k) Frequency shifting e s(t) S(k - m) * e 2π k t j S(k)
25 FS - oddities FS - oddities Orthonormal base Fourier components {uk}{ } form orthonormal base of signal space: u k = (/ ) exp(jkωt) ( k =, 2, + ) Def.: Internal product : u k u m = δ k,m ( if k = m, otherwise). (Remember (e jt )* = e -jt ) u k u m = u u o k * m dt hen c k = (/ ) s(t) u k i.e. (/ ) times projection of signal s(t) on component u k Negative frequencies & time reversal k = -, -2,-,,,2, +, ω k = kω, φ k = ω k t, phasor turns anti-clockwise. Negative k phasor turns clockwise (negative phase φ k ), equivalent to negative time t, time reversal. Careful: phases important when combining several signals!
26 FS - power FS - power Average power W : Parseval s s heorem 2 W = = 2 ck a + k= 2 W = o s(t) ak bk k= dt s(t) s(t) FS convergence ~/k lower frequency terms W k = c k 2 carry most power. W k vs. ω k : Power density spectrum. Example Pulse train, duty cycle δ = 2 τ / s(t) 2 τ 2 - W k /W W k = 2 W sync 2 (k δ) -2 b k = a = δ s MAX a k = 2δs MAX sync(k δ) t -3 W = (δ s MAX ) sync(u) = sin(π u)/(π u) W W = + k W W k= k f
27 FS of main waveforms FS of main waveforms
28 Discrete Fourier Series (DFS) Discrete Fourier Series (DFS) Band-limited signal s[n], period = N. DFS defined as: N 2πk n j c ~ k = s[n] e N N n= ~ ~ Note: c k+n = c k same period N analysis analysis i.e. time periodicity propagates to frequencies! synthesis synthesis N 2πk n j s[n] = c ~ k e N k= Synthesis: finite sum band-limited s[n] DFS generate periodic c k with same signal period Orthogonality in DFS: N N 2π n(k-m) j e N n= = δ k,m Kronecker s delta N consecutive samples of s[n] completely describe s in time or frequency domains.
29 DFS analysis DFS analysis DFS of periodic discrete -Volt square-wave s[n] c ~ k s[n]: period N, duty factor L/N L, k =, + N, ± 2N,... N = πk(l ) j π kl sin e N N, otherwise N π k sin N Discrete signals periodic frequency spectra. Compare to continuous rectangular function (slide #, FS analysis - ) amplitude amplitude.24 phase phase.2π n L N.6.6 c k.4π k θ k n -.4π π ~.2π.6.6.4π -.4π π
30 DFS properties DFS properties ime 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 * Convolution * N m= s[n] u[n] s[m] u[n m] N h= S(k) U(k) ime shifting s[n - m] e S(k) Frequency shifting 2π h t + j e s[n] S(k - h) N S(h)U(k -h) 2π k m j
31 DF Window characteristics DF Window characteristics Finite discrete sequence spectrum convoluted with rectangular window spectrum. Leakage amount depends on chosen window & on how signal fits into the window. () Resolution: capability to distinguish different tones. Inversely proportional to mainlobe width. Wish: as high as possible. (2) (3) Peak-sidelobe level: maximum response outside the main lobe. Determines if small signals are hidden by nearby stronger ones. Wish: () as low as possible. (2) Sidelobe roll-off: sidelobe decay per decade. rade-off with (2). (3) Several windows used (application( application- dependent): Hamming, Hanning, Blackman, Kaiser... Rectangular window
32 DF of main windows DF of main windows Windowing reduces leakage by minimising sidelobes magnitude. In time it reduces endpoints discontinuities. Sampled sequence Non windowed Windowed Some window functions
33 DF - Window choice DF - Window choice Common windows characteristics Window type -3 db Mainlobe width -6 db Mainlobe width Max sidelobe level Sidelobe roll-off [db/decade] [bins] [bins] [db] Rectangular Hamming Hanning Blackman Observed signal Far & strong interfering components Near & strong interfering components Accuracy measure of single tone Window wish list high roll-off rate. small max sidelobe level. wide main-lobe NB: Strong DC component can shadow nearby small signals. Remove it!
34 DF - Window loss remedial DF - Window loss remedial Smooth data-tapering tapering windows cause information loss near edges. Solution: sliding (overlapping) DFs. 2 x N samples (input signal) DF # Attenuated inputs get next window s full gain & leakage reduced. Usually 5% or 75% overlap (depends on main lobe width). DF #2 DF #3 Drawback: increased total processing time. DF AVERAGING
35 DF - parabolic interpolation DF - parabolic interpolation Rectangular window Hanning window Parabolic interpolation often enough to find position of peak (i.e. frequency). Other algorithms available depending on data.
36 Systems spectral analysis (hints) Systems spectral analysis (hints) System analysis: measure input-output relationship. Linear ime Invariant x[n] DIGIAL LI SYSEM h[n] y[n] δ[n] n DIGIAL LI SYSEM h[n] h[t] = impulse response n x[n] h[n] y[n] = x[n] h[n] = m= x[n m] h[m] y[n] predicted from { x[n], h[t] } X(f) H(f) Y(f) = X(f) H(f) H(f) ) : LI transfer function ransfer function can be estimated by Y(f) / X(f)
37 Estimating H(f) (hints) Estimating H(f) (hints) G G xx yx * (f) = X(f) X (f) Power Spectral Density of x[t] (F of autocorrelation). * (f) = Y(f) X (f) Cross Power Spectrum of x[t] & y[t] (F of cross-correlation). * Y(f) Y(f) X (f) Gyx = = ransfer Function X(f) * X(f) X (f) Gxx (ex: beam!) H(f) = It is a check on H(f) validity!
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