23 April 2014 ECE 438 Essential Definitions and Relations CT Metrics 4. x(t)y(t) 1. Energy Ex 2 x(t) dt x(t )y()d x(t kT ) 1 2 x(t) dt 2. Power Px lim T T T /2 5. rep T [x(t)] 3. root mean squared value xrms Px 6. comb T [x(t)] T /2 k x(kT) (t kT) k 4. Area Ax x(t)dt Properties of the CT Impulse T /2 1 x(t)dt T T T /2 5. Average value xavg lim 1. 6. Magnitude Mx max x(t) 2. 1. Energy Ex x[n] 2 n N 1 2 2. Power Px lim x[n] N 2N 1 n N 3. root mean squared value xrms Px 4. Area Ax x[n] and (t) 0, t 0 t DT Metrics (t)dt 1 (t t0 ) x(t)dt x(t 0 ), provided x(t) is continuous at t t 0 3. x(t) (t t 0 ) x(t 0 ) (t t0 ) 4. (t t0 ) x(t) x(t t0 ) 1 5. (at t 0 ) a (t t 0 / a) n N 1 x[n] N 2N 1 n N 5. Average xavg lim 6. Magnitude Mx max x[n] n CT Signals and Operators 1, | t|< 1 / 2 1. rect (t) = 0, | t|> 1 / 2 CTFT Sufficient conditions for existence |x(t )|dt | x(t)| 2 or Forward transform X( f ) x(t)e j 2 ft dt 2. sinc (t) = sin(t) t 1 t 3. (t ) lim rect 0 Inverse transform x(t) X( f )e j 2 ft df dt 23 April 2014 -2 - Transform Relations ECE 438 Essential Definitions ... CT FT 1. rect (t) sinc (f) 1. Linearity CT FT a1 x1(t) a2 x 2 (t) a1X1( f ) a2 X 2 ( f ) 2. Scaling and shifting t t0 CT FT j2ft 0 x |a| X(af )e a j2 f 0 t CT FT CT FT 3. e j2 f 0 t ( f f 0 ) CT FT 1 4. 3. Modulation x(t)e CT FT 2. (t ) 1 and 1 ( f ) cos(2f 0 t) 2 {( f f 0 ) ( f f 0 )} CT FT X( f f 0 ) DT Signals and Operators 4. Reciprocity 1, n 0 1. u[n] = 0, n < 0 CT FT X(t) x( f ) 5. Parseval |x(t )| 2 dt | X( f )| 2 df 1, 2. [n] = 0, 3. x[n]y[n] 6. Initial Value x[n k]y[k] x(t) dt x(0) n0 k X(0) n0 X( f ) df x[n] CT FT x(t)y(t) X( f )Y( f ) y[n] 4. Downsampling y[n] x[Dn] 7. Convolution D x[n] D y[n] 5. Upsampling x[n / D], n / D integer - valued y[n] else 0, 8. Product CT FT x(t)y(t) X ( f )Y( f ) 9. Transform of periodic signal CT FT 1 rep T [x(t)] comb 1 [X( f )] T T 10. Transform of sampled signal CT FT 1 comb T [x(t)] rep 1 [X( f )] T T Properties of the DT Impulse 1. [n] 1 and [n] 0, n 0 n 2. [n n0 ] x[n] x[n0 ] n 3. [n n0 ]x[n] x[n n0 ] DTFT Transform Pairs Sufficient conditions for existence 23 April 2014 -3 - ECE 438 Essential Definitions ... x[n] 2 x[n] or n n Forward transform x[n] 8. Downsampling 1 D1 2k Y () X D k 0 D x[n]e jn X() n y[n] D x[n] y[n] D 10. Upsampling Y() X(D) Inverse transform 1 jn x[n] X()e d 2 Transform Pairs DTFT 1. [n] 1 Transform Relations DT FT 2. 1 2 rep2 [()] 1. Linearity DT FT a1 x1[n] a2 x 2 [n] a1X1 () a2 X 2 () DT FT j n 3. e 0 2 rep 2 [( 0 )] 2. Shifting DTFT x[n n0 ] X( )e jn 0 3. Modulation x[n]e j0 n DTFT X( 0 ) 4. Parseval 1 2 x[n] 2 X() d n 2 5. Initial Value X(0) x[n] DT FT 4. cos( 0 n) rep 2 [( 0 ) ( 0 )] 1, n 0,1,..., N 1 DTFT x[n] else 0, 5. sin( N / 2) j ( N 1) / 2 X( ) e sin( / 2) Relation between CT and DT 1. Time Domain xd [n] x a (nT) n 1 x[0] X()d 2 2. Frequency Domain Xd ( d ) f s rep f s X a ( f a ) where f s 6. Convolution DT FT x[n]y[n] X()Y() 1 T ZT 7. Product DT FT 1 x[n]y[n] X ( )Y ()d 2 Forward transform X(z) x[n]z n n fa d f s 2 , 23 April 2014 -4 - Transform Relations ECE 438 Essential Definitions ... N1 X[k] 1. Linearity j2 kn/ N n0 ZT a1 x1[n] a2 x 2 [n] a1 X1 (z) a2 X2 (z) 2. Shifting ZT x[n n0 ] X( )z n0 3. Modulation n x[n]z0 x[n]e Inverse transform 1 x[n] N N 1 X[k]e j2 kn /N k 0 Transform Relations ZT X(z / z0 ) 1. Linearity DFT 4. Multiplication by time index ZT d n x[n] z X(z) dz a1 x1[n] a2 x 2 [n] a1X1[k] a2 X2 [k] 2. Shifting DFT x[n n0 ] X[k ]e 6. Convolution ZT x[n]y[n] X(z)Y(z) 7. Relation to DTFT If X ZT(z) converges for |z| 1 , then the DTFT of x[n] exists and XDTFT( ) XZT(z) ze j j2 n 0k / N 3. Modulation DFT x[n]e j2nk0 / N X[k k0 ] 4. Reciprocity DFT X[n] N x[k] 5. Parseval Transform Pairs N1 ZT 1. [n]1, all z ZT 2. a nu[n] ZT ZT 4. n a u[n] ZT 5. na u[n 1] n DFT Forward transform k0 n 0 x[0] 2 2 N 1 1 1 az 1 X[k ] X[0] x[n] 1 1 , |z||a| 1 az az N 1 6. Initial Value 1 1 , |z||a| 1 az 3. anu[n 1] n n 0 1 x[n] N 2 , | z|| a| 1 N 1 X[k] N k0 7. Periodicity az 1 1 az 1 2 , |z||a| x[n mN] x[n] for all integers m X[k lN ] X[k] for all integers l 8. Relation to DTFT of a finite length sequence Let x0 [n] 0 only for 0 n N 1 23 April 2014 -5 - ECE 438 Essential Definitions ... DTFT b x0 [n] X0 ( ) . P{a X b} f X (x)dx Define x[n] a x [n mN] , 0 f X (x)dx 1 m x[n] X[k] , X[k] X0 e then j2 k /N f X (x) 0 DFT 2. Probability distribution function FX (x) . x f X ( )d FX (x) 6. Periodic convolution DFT x[n] y[n] X[k]Y[k] f X (x) 7. Product DFT x[n]y[n] 1 X[k] Y [k] N dFX (x) dx FX () 0 FX () 1 FX (x) is a non-decreasing function of x Expectation Operator Transform Pairs E{g(X)} DFT 1. [n], 0 n N 1 1. Mean g(x) x 1, 0 k N 1 2. 2nd moment g(x) x 2 DFT 2. 1, 0 n N 1 N [k], 0 k N 1 DFT 3. e j2 k 0 n /N , 0 n N 1 N [k - k 0 ], 0 k N 1 DFT cos(2 k0 n / N ), 0 n N 1 4. N [k - k 0 ] + [k - (N - k0 )], 2 0 k N 1 j 0 n DFT sin (2k / N 0 ) / 2 4. Relation between mean, 2nd moment, and variance 2X E{X2} (E{X})2 Two Random Variables 1. Bivariate probability density function f XY (x, y) bd P{a X b,c Y d} f XY (x,y)dxdy ac f XY (x, y) 0 ,n 0,1,..., N 1 sin (2k / N 0 )(N / 2) 2 3. Variance g(x) (x E{X}) f XY (x,y)dxdy 1 5. e g(x) f X (x)dx e j( 2 k / N 0 )( N 1 ) /2 Random Signals 1. Probability density function f X (x) 2. Marginal probability density functions f X (x) and f Y (y) f X (x) f XY (x, y)dy 23 April 2014 -6 f Y (y) f XY (x, y)dx 3. Linearity of expectation E{ag(X,Y) bh(X,Y)} aE{g(X,Y)} bE{h(X,Y)} 4. Correlation of X and Y E{XY} xyf XY (x, y )dxdy 5. Covariance of X and Y 2XY E{(X E{X})(Y E{Y})} E{XY} E{X}E{Y} 6. Correlation coefficient of X and Y XY 2XY X Y 0 | XY |1 7. X and Y are independent if and only if f XY (x, y) f X (x) f Y (y) 8. X and Y are uncorrelated if and only if E{XY} E{X}E{Y} Independence implies uncorrelatedness. The converse is not true. Discrete-time random signals 1. Autocorrelation function r XX[m,m n] E{X[m]X[m n]} X[m] is wide-sense stationary if and only if mean is constant, and autocorrelation depends only on difference between arguments, i.e. E{X[n]} X r XX[n] E{X[m]X[m n]} 2. Cross-correlation function r XY[m, m n] E{X[m]Y[m n]} ECE 438 Essential Definitions ... X[m] and Y[m] are jointly wide-sense stationary if and only if they are individually wide-sense stationary and their cross-correlation depends only on difference between arguments, i.e. r XY[n] E{X[m]Y[m n]} 3. Wide-sense stationary processes and linear systems. Let X[n] be wide-sense stationary, and suppose that Y[n] h[n] X[n] h[n m]X[m] , m then X[n] and Y[n] are jointly widesense stationary, and Y X h[m] m rXY [n] h[n] rXX [n] rYY [n] h[n] rXY [n] Model for Speech Our model for the speech waveform is given by s[n] v[n]e[n] , where s[n] is the speech waveform, v[n] is the vocal tract response that typically contains one or more peaks corresponding to resonant frequencies, and e[n] is the excitation provided by the glottis. [n kP], for a voiced phoneme, e[n] k white noise, for an unvoiced phoneme. Here P is the pitch period in samples. Short-Time Discrete Time Fourier Transform (STDTFT) Two interpretations: 23 April 2014 -7 - 1. DTFT of windowed waveform centered at (fixed) n where w[n] is the window function S( , n) s[k]w[n k]e ECE 438 Essential Definitions ... that the baseband filter impulse response h0 [n] satisfy 1 , n 0 . h0 [n] L 0, n kL j k k 2. Response of narrowband filter centered at (fixed) frequency , after shifting to base-band. The impulse response of the narrowband filter is h [n] h0 [n]e j n . Here h0 [n] is the impulse response of the base-band filter, which is equivalent to the window function w[n] in the first interpretation. We thus have S( , n) h [n] s[n]e j n The display of the 2-D image S( ,n) as a function of and n is called a spectrogram. Let N be the length of the window function w[n] ; and let P be the pitch period. If N ? P , the spectrogram is narrowband. If N P , the spectrogram is wideband. There are no constraints on h0 [n] for other values of n . Miscellaneous 1. Geometric Series N1 zn n 0 2. N roots zk , k 0, , N 1 , of a complex constant u0 , i.e. N solutions to N z u0 0 zk u0 1/ N e cos(2 ) cos 2 sin 2 Consider an L-channel modulated filter bank with input x[n] and output from the -th channel given by cos 2 sin sin cos cos hl [n] h0 [n]e j 2 n / L , l 0, ..., L 1 . The output of the filter bank is formed by summing the outputs of all L channels L 1 y[n] yl [n] l 0 A necessary and sufficient condition for perfect reconstruction, i.e. y[n] x[n] is , k 0, ,N 1 sin(2 ) 2 sin cos sin 2 where j 2 k u 0 / N 3. Trigonometric Identities sin( ) sin cos cos sin cos( ) cos cos msin sin Modulated Filter Bank yl [n] hl [n] x[n] , 1 z N 1 z sin cos cos sin 1 1 cos(2 ) 2 2 1 1 cos(2 ) 2 2 1 cos( ) cos( ) 2 1 cos( ) cos( ) 2 1 sin( ) sin( ) 2 1 sin( ) sin( ) 2