Signal Processing Summary

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1 Signal Processing Summary Jan Černocký, Valentina Hubeika DCGM FIT BUT Brno, FIT BUT Brno Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 1/41

2 Agenda Introduction Analog signals and sampling Frequency analysis of sampled signals. Random signals. Filters. Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 2/41

3 Why digital signal processing? reproducibility (we don t have to think about component tolerance ). no changes due to material aging or temperature. easy setup (may be difficult tuning for analog signals). possible adaptive processing ( changing device functionality depending on the type of the signal ). simulation = application. Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 3/41

4 Usual Approach x(t) A/D x (n) digital processing y (n) D/A y(t) (PC, DSP). storing transmision interpretation other processing... The input signal is continuous: function of a real-valued variable (in our case time) t, t is defined over the interval (, ), t. Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 4/41

5 s(t) t To represent the signal in frequency domain (spectrum), Fourier transform is used: X(f) = x(t)e j2πft dt, (1) where the function X(f) is called the spectral function, or in short spectrum. The function is defined f from to and is complex. Its magnitude is X(f) and its phase is X(f). Here, we talk about magnitude and phase spectra. Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 5/41

6 In case of real signals, we will need only the right side of the spectral function (f > ). The left side is complex adjoint to the right side: or X(f) = X( f) a arg X(f) = arg X( f). X(f) = X ( f), (2) S(f) -f max f f max Good signals are frequency constrained (energy within the band of (, f max ) ). Spectral function cant be calculated (infinite, integral,...). Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 6/41

7 Analog to Digital (AD) Converter x(t) antialias. filter sampling sampled signal xs(n) quantization quantized signal xq(n) Sampled signal is calculated by multiplying the original signal with a periodic function over time. s(t) D s(t) D x (t)=x(t)s(t) s 1/D T t t Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 7/41

8 Theoretically, sampling is multiplying the original analog signal with a series od Dirac deltas (unit impulse function) (infinite height, zero width, unit area). The result of multiplication is again a series of Dirac impulses, with the powers corresponding to the values of the original signal at times nt: s(t) s(t) 1 D 1 x (t)=x(t)s(t) s T t t T is the sampling period F s = 1 T is the sampling frequency Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 8/41

9 The spectrum of the sampled signal is periodized!!! X s (f) = 1 T + X ( f n T ) = 1 T + X (f nf s ) (3) n= n= From the point of view of the sampling theorem: 1) F s > 2f max : The copies of the spectrum of the original signal do not overlap so the original signal can be perfectly reconstructed from its sampled version by filtering it with low-pass filter with the frequency F s /2. 2) F s 2f max : the copies of the spectrum of the original signal do overlap so the resulting spectrum has a different shape than the original spectrum. The original signal cannot be reconstructed from its sampled version. We observe so called aliasing. Shanon Kotelnikov Nyquist sampling theorem F s > 2f max Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 9/41

10 1. Example: sampling and reconstruction OK F s = 8 Hz, f max = 3 Hz, so that Ω s = 16π rad/s, ω max = 6π rad/s. T = 1 8 s x x x x x 1 5 Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 1/41

11 2. Example: sampling and reconstruction BAD F s = 8 Hz, f max = 7 Hz, so that Ω s = 16π rad/s, ω max = 14π rad/s. T = 1 8 s x x x x x 1 5 Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 11/41

12 Antialiasing filter constrain to [ F s /2, F s /2] x x x x x x 1 5 Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 12/41

13 Denotation of the Sampled Signal x s (nt) or just x s [n] the sampled signal is a series of values. 1. Along with the sampled signal, we always need information on the sampling frequency (implicit or explicit) 2. When working with a sampled signal, we do not want to consider the sampling frequency in the calculations, therefore assume that the period is T = 1, thus F s = 1. The normalized time is then given as: The normalized frequency is: t = t T, thus n = nt T (4) f = f F s (5) Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 13/41

14 Example Define a function generating cosine with the frequency of 2 Hz with the sampling frequency F s = 8 Hz. In the continuous time: s(t) = cos(2πf t) = cos(2π2t). While sampling we will substitute the continuous time variable( t with the discrete time nt, where T is the sampling period: x(nt) = cos(2πf nt) = cos 2π f ) n. F s The frequency f ( is normalized. The desired signal is defined as: x(n) = cos 2π 1 ) F s 4 n. Generate 1 second of such a signal in Matlab: n = :7999; x = cos (2 * pi * 1 / 4 * n); wavwrite(x,8,16, sig.wav ); Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 14/41

15 Behaviour of the Sampled Signal in Frequency Domain Spectrum Discrete Fourier transform DTFT definition: X(k) = N 1 n= nk j2π x[n]e N pro k <, N 1 > (6) How to use it for discrete signals?: analyze a window of the length of N samples. result: when multiplying values X(k) with the sampling perioud T, the approximation of the spectral function at the points k f, where f = F s N f = 1 N (normalized frequency (usually not used) (true frequency) or ˆX(k f) = T N 1 n= nk j2π x[n]e N (7) in this definition, x[n] can be substitute for x(nt). Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 15/41

16 Comparing to the spectral function of the corresponding analog signal, the spectrum of the discrete signal is not the same thing! 1. we calculate spectrum of the sampled signal, so the spectrum is periodic with the period N units (corresponds to the sampling frequency F s ). If k (, + ), then ˆX(k f) is repeated after each N values. 2. The signal is windowed. The resulting spectrum carries information on the window: the signal is multiplied with the window function in time, the spectrum of the window is then convoluted with the spectrum of the original signal. This often results in blurring of the theoretically sharp lines of the spectrum (for instance during analysis of the harmonic signal). More in the lecture on preprocessing. 3. if the spectrum is discrete (we dispose of only N values from to F s ), then the signal must be periodic! time sampling periodization frequency periodization discretization Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 16/41

17 How can we use it in practice? We want to make frequency analysis of one voiced frame of speech s = wavread( test.wav ) ; sfr = frame (s,16,8); x = sfr(:,13); plot (x); Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 17/41

18 DTFT only careful with the frequency axes: Fs = 8; f = (:159) / 16 * Fs; X = fft(x); subplot (211); plot(f,abs(x)); subplot (212); plot(f,angle(x)); Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 18/41

19 DTFT only the left half is symmetric to the right half (we consider only on in analysis) Fs = 8; f = (:79) / 16 * Fs; X = fft(x); X = X(1:8); subplot (211); plot(f,abs(x)); subplot (212); plot(f,angle(x)); Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 19/41

20 If we want to smoother the spectrum (add more points shorten intervals between values), having only one frame (without a possibility of extension), we can use zero padding Fs = 8; f = (:511) / 124 * Fs; X = fft([x zeros(1,124-16)]); X = X(1:512); subplot (211); plot(f,abs(x)); subplot (212); plot(f,angle(x)); Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 2/41

21 FREQUENCY ANALYSIS OF RANDOM SIGNALS Speech signal is a random signal. Thus we use power spectral density (PSD) function for its frequency analysis. PSD is real and represent power density of the signal in frequency domain. One of the possible ways of computation uses DTFT: Which is just the power of the magnitude. Ĝ DTFT (k f) = 1 N X[k] 2. careful with X[k] 2. in Matlab power of the complex number is not the same as the power of its magnitude: X = fft(x); Gdft = X.^ 2; wrong! X = fft(x); Gdft = abs(x).^ 2; good! X = fft(x); Gdft = X.* conj(x); good and fast Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 21/41

22 Example: Fs = 8; f = (:511) / 124 * Fs X = fft([x zeros(1,124-16)]); X = X(1:512); Gdft= 1/16 *abs(x).^ 2; plot(f,gdft); Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 22/41

23 Dynamic performance of the power spectral density is higher than of DTFT (second power) which causes poor visualization of weak parts of the image. For this reason, often decibels are used (Matlab: function log1) Ĝ DTFT (k f) = 1 log 1 1 N X[k] Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 23/41

24 LINEAR FILTERS x(n) filter y(n) Usually, filters are linear hold linear combination : if x 1 (n) y 1 (n) a x 2 (n) y 2 (n), then ax 1 (n) + bx 2 (n) ay 1 (n) + by 2 (n), where a, b R. time invariant behaviour does not depend on time: if x(n) y(n), then as well x(n n ) y ( n n ), where n is any arbitrary shift. Sometimes however we desire filters with time variant behaviour adaptive systems, speech frames (change 1 ms). causal the filter does not see into future : y(n) y(m < n) a x(m n). Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 24/41

25 Impulse Response or impulse characteristics is the response of a filter to the Kroneck or unit impulse (do not confuse with the Dirac delta): pro n δ(n) = (8) 1 pro n = δ(n) h(n) filter Knowing the impulse response, we can calculate the response to any input signal. Each sample is perceived as a shifted and scaled unit impulse. We represent the output as the convolution: y(n) = x(n) h(n) = m= x(m)h(n m) = m= h(m)x(n m) (9) Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 25/41

26 Analysis of the impulse characteristics: if h(k) = for k <, then the filter is causal (all samples after n-th sample will be multiplied by zero). impulse response can be finite FIR (finite impulse response) or infinite IIR (infinite impulse response). the Fourier image in frequency then represents the complex characteristics of the filter: h(k) H(f) Convolution in time domain correspond to product in frequency domain, thus the spectrum of the resulting signal is: Y (f) = X(f)H(f) (1) We should keep in mind that we work with the discrete signals (so is the impulse response of a filter) the image in the frequency is therefore periodic with the period F s or 1 for the normalized frequency. Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 26/41

27 Representation of a Filter y(n) x(n) z -1 z z -1 b Q Σ... z -1 z -1 z -1 b Q-1 b Q-2 -a -a b 1 -a P-1 b -a P The block z 1 denotes a delay of 1 sample. The behaviour of a filter can be expressed as a difference equation: y(n) = Q b k x(n k) k= P a k y(n k), (11) Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 27/41 k=1

28 where x(n k) are actual and delayed input values and y(n k) are delayed output values. Types of filters: FIR non-recursive: only b...b Q nonzero. Always stable. IIR solely recursive: only b, a 1...a P nonzero. IIR generally recursive: a i and b i nonzero. It is difficult to estimate the behaviour of the filter and analyze it from the point of view of stability from the difference equation. Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 28/41

29 Z-Transform X(z) = n= x(n)z n (12) The z-transform is a generalization of the DTFT. The z-transform becomes the DTFT when z = exp jw (when DTFT is evaluated on the unit circle). In order to determine the frequency response of the system the z-transform must be evaluated on the circle (the region of convergence must contain a circle). Otherwise, the DTFT of the system does not exist. Lets define delay as: if x(n) X(z), then for y(n) = x(n n ) will hold: Y (z) = z n X(z) (13) for the delay of one time unit holds: x(n 1) z 1 X(z). Thus the delay of one time unit will be denoted as: z -1 Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 29/41

30 Transfer Function of a Filter We can rewrite the difference equation using z-transform Y (z) = Q b k X(z)z k k= P a k Y (z)z k, (14) k=1 The transfer function can be defined as the fraction: H(z) = Y (z) X(z) = 1 + Q b k z k k= = P a k z k k=1 B(z) A(z), (15) where B(z) and A(z) are two polynomes. The a coefficient of the denominator must be equal to 1. a does not physically present in the filter (denotes that the filter has the output). Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 3/41

31 Frequency Characteristics of a filter from to F s (or from to 1 in using normalized frequency) can be acquired from the transfer function by analyzing the unit circle. We are interested in the complex values of the function H(z): for the normalized frequency. Or: H(f) = H(z) z=e j2πf (16) H(f) = H(z) z=e j2πft (17) for the normal frequency. For each value f we need to calculate the position of the point on the unit circle: z = e j2πf (complex number), then for this number we need to compute the fraction of the polynomes B(z) and A(z) (complex number). In Matlab by using function freqz. Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 31/41

32 +j Im "z" exp(j2 πf) 1 1/2~Fs/2 1 1~Fs Re j Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 32/41

33 Zeros and Poles of the Transmission Function... The other way how to represent the transfer function H(z) is using product: H(z) = B(z) A(z) = b + b 1 z b Q z Q 1 + a 1 z a P z P = z Q (b z Q + b 1 z Q b Q ) z P (z P + a 1 z P a P ) = = b z Q z P Q (z n k ) k=1 P (z p k ) k=1 = b z P Q Q (z n k ) k=1 P (z p k ), If a k, b k R, then poles p k and zeros n k can be either real or complex conjugate in pairs. Filter is stable when all poles lie within the unity circle: k=1 p k < 1 Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 33/41

34 Example of a Filter We want to implement a filter that will be simulating a telephone channel to filter a signal of the CD quality. Our filter will be a band-pass filter from 3 to 34 Hz. There are several functions available in Matlab to implement a filter. We will be using so called elliptic filters: Fs = 441; Fs2 = Fs/2; % has to be normalized by half Fs Wp = [3/Fs2 34/Fs2]; % pass-band Ws = [2/Fs2 35/Fs2]; % stop-band - approximately Rp = 3; % ripple in pass-bandu db Rs = 3; % muting stop-bandu db (obe hodnoty od % see the norms. [N, Wn] = ellipord(wp, Ws, Rp, Rs) % computing of the filter order [B,A] = ellip(n,rp,rs,wn) % computing of polynoms B a A...as the result we have 2 polynomes of the 12th order. Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 34/41

35 Frequency characteristics: freqz (B,A,512,Fs); 5 Magnitude (db) Frequency (Hz) x 1 4 Phase (degrees) Frequency (Hz) x 1 4 Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 35/41

36 Poles and zeros: zplane (B,A); Imaginary Part Real Part Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 36/41

37 Implementation in C the basic implementation of the direct structure is very simple just rewrite the difference equation into C (see the file filter.c) in practice, usually optimal structures are used, which have only one delay bus and are not prone to the rounding errors. for more information on the filter theory see SXC/ISS lecture discrete systems : Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 37/41

38 Passing a Random Signal through a Filter Let the filter have a complex frequency characteristic H(f). For the input signal with the spectral power density G x (f) the output spectral power density is given as: G y (f) = H(f) 2 G x (f)...the input PSD is multiplied by the power of the magnitude of the complex frequency characteristic. Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 38/41

39 Example: Given a realization of a random signal. The signal is filtered by the filter defined as H(z) = 1.9z 1. The output ignal and its PSD are: Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 39/41

40 The magnitude of the frequency characteristic and its power are: H H Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 4/41

41 The output signal and its PSD: Signal Processing Summary Jan Černocký, Valentina Hubeika, CDGM FIT BUT Brno 41/41

Signal Processing for Speech Applications - Part 2-1. Signal Processing For Speech Applications - Part 2

Signal Processing for Speech Applications - Part 2-1 Signal Processing For Speech Applications - Part 2 May 14, 2013 Signal Processing for Speech Applications - Part 2-2 References Huang et al., Chapter