EE 438 Essential Definitions and Relations

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1 May 2004 EE 438 Essential Definitions and Relations CT Metrics. Energy E x = x(t) 2 dt 2. Power P x = lim T 2T T / 2 T / 2 x(t) 2 dt 3. root mean squared value x rms = P x 4. Area A x = x(t) dt 5. Average value T / 2 x avg = T lim x(t ) 2T dt T / 2 6. Magnitude M x = max x(t) <t < DT Metrics. Energy E x = 2 n = 2. Power P x = lim N 2N + N n = N 2 3. root mean squared value x rms = P x 4. Area A x = n = 5. Average x avg = lim N 2N + 6. Magnitude M x = max <n < CT Signals and Operators, t < / 2. rect (t) = 0, t > / 2 2. sinc(t) = sin(πt) πt 3. δ(t ) = lim Δ 0 Δ rect t Δ N n = N 4. x(t) y(t) = x(t τ)y( τ)dτ 5. rep T [x(t)] = x(t kt ) k = 6. comb T [x(t)] = x(kt) δ(t kt) k = Properties of the CT Impulse. δ(t) dt = and δ(t) = 0, t 0 2. δ(t t 0 ) x(t)dt = x(t 0 ), provided x(t) is continuous at t = t 0 3. x(t)δ(t t 0 ) x(t 0 )δ(t t 0 ) 4. δ(t t 0 ) x(t) = x(t t 0 ) 5. δ(at t 0) a δ(t t 0 / a) CTFT Sufficient conditions for existence x(t ) dt < or x(t) 2 dt Inverse transform X( f ) = x(t)e j 2πft dt x(t) = X( f )e j 2πft df

2 May EE 438 Essential Definitions.... Linearity a x (t) + a 2 x 2 (t) CTFT a X ( f ) + a 2 X 2 ( f ) 2. Scaling and shifting x t t 0 CTFT a a X(af )e j2πft 0 x(t)e j2 πf 0 t CTFT X( f f 0 ) 4. Reciprocity X(t) CTFT x( f ) 5. Parseval x(t ) 2 dt = X( f ) 2 df 6. Initial Value X(0) = x(t) dt x(0) = X( f ) df 7. Convolution x(t) y(t) CTFT X( f )Y( f ) 8. Product x(t)y(t) CTFT X ( f ) Y( f ) 9. Transform of periodic signal CTFT rep T [x(t)] T comb [X( f )] T 0. Transform of sampled signal CTFT comb T [x(t)] T rep [X( f )] T. rect (t) CTFT sinc(f) 2. δ(t ) CTFT and CTFT δ( f ) 3. e j2πf t CTFT 0 δ( f f 0 ) 4. cos(2πf 0 t) CTFT 2 {δ( f f 0 ) DT Signals and Operators. u[n] =, n 0 0, n < 0 2. δ[n] =, n = 0 0, n 0 + δ( f + f 0 )} 3. y[n] = x[n k]y[k] k = D 4. Downsampling y[n] = x[dn] y[n] y[n] D 5. Upsampling x[n / D], n / D integer - valued y[n] = 0, else Properties of the DT Impulse. δ[n] = and δ[n] = 0, n 0 n= 2. δ[n n 0 ] = x[n 0 ] n = 3. δ[n n 0 ] = x[n n 0 ] DTFT Sufficient conditions for existence

3 May EE 438 Essential Definitions... < or 2 < n = X(ω) = Inverse transform = 2π n= π π n = e jωn X(ω)e jωn dω. Linearity a x [n] + a 2 x 2 [n] DTFT a X (ω) + a 2 X 2 (ω) 2. Shifting x[n n 0 ] DTFT X(ω)e jωn 0 4. Parseval e jω 0 n DTFT X(ω ω 0 ) 2 = 2π n = 5. Initial Value X(0) = n= x[0] = 2π π π π π X(ω) 2 dω X(ω)d ω 6. Convolution y[n] DTFT X(ω)Y(ω) 7. Product y[n] DTFT 2π π X (ω µ)y(µ)d µ π 8. Downsampling Y(ω) = D D k= 0 D y[n] X ω 2πk D y[n] D 0. Upsampling Y(ω) = X(Dω). δ[n] DTFT 2. DTFT 2π rep 2π [δ(ω)] 3. e jω n DTFT 0 2π rep 2 π [δ(ω ω 0 )] 4. cos(ω 0 n) DTFT π rep 2π [δ(ω ω 0 ) + δ(ω + ω 0 )] 5. =, n = 0,,..., N 0, else X(ω) = DTFT sin(ωn / 2) sin(ω / 2) e jω ( N ) / 2 Relation between CT and DT. Time Domain x d [n] = x a (nt) 2. Frequency Domain ZT [ ] f a = ω d f s 2π X d (ω d ) = f s rep f s X a ( f a ) X(z) = where f s = T n = z n,

4 May EE 438 Essential Definitions.... Linearity a x [n] + a 2 x 2 [n] ZT a X (z) + a 2 X 2 (z) 2. Shifting x[n n 0 ] ZT X(ω )z n 0 z 0 n ZT X(z / z 0 ) 4. Multiplication by time index 6. Convolution n ZT z d dz [ X(z) ] y[n] ZT X(z)Y(z) 7. Relation to DTFT If X ZT (z) converges for z =, then the DTFT of exists and X DTFT (ω ) = X ZT (z) z=e jω. δ[n] ZT, all z 2. a n u[n] ZT, z > a az 3. a n u[ n ] ZT az, z < a 4. n a n u[n] ZT az ( az 2, z > a ) 5. na n u[ n ] ZT az ( az ) 2, z < a DFT X[k] = N n=0 Inverse transform = N N k =0. Linearity j2 πkn/ N e j2 πkn /N X[k]e a x [n] + a 2 x 2 [n] DFT a X [k] + a 2 X 2 [k] 2. Shifting 4. Reciprocity 5. Parseval x[n n 0 ] DFT X[k ]e j2πn 0 k / N e j2πnk 0 / N DFT X[k k 0 ] N n=0 6. Initial Value 7. Periodicity X[n] DFT N x[ k] 2 = N X[0] = N n =0 x[0] = N x[n + mn] = X[k + ln ] = X[k] N k= 0 N k = 0 X[k] X[k ] 2 for all integers m for all integers l 8. Relation to DTFT of a finite length sequence Let x 0 [n] 0 only for 0 n N

5 May EE 438 Essential Definitions... Define = x 0 [n] DTFT X 0 (ω). x 0 [n + mn], m= DFT X[k], ( ). j2 πk /N then X[k] = X 0 e 6. Periodic convolution 7. Product y[n] DFT X[k]Y[k] y[n] DFT X[k ] Y[k] δ[n], 0 n N DFT., 0 k N, 0 n N DFT 2. Nδ[k], 0 k N e j2 πk 0 n/n, 0 n N DFT 3. Nδ[k - k 0 ], 0 k N cos(2πk 0 n / N ), 0 n N DFT N { 2 δ[k - k 0 ] + δ[k - (N- k 0 )] }, 0 k N e jω 0 n,n = 0,,..., N DFT [ ] sin [(2πk / N ω 0 ) / 2] sin (2πk / N ω 0 )(N / 2) Random Signals j( 2 πk / N ω 0 )( N ) /2 e. Probability density function f X (x) P{a < X b} = b a f X (x)dx f X (x)dx = f X (x) 0 2. Probability distribution function F X (x) F X (x) = x f X (ξ )dξ f X (x) = df X (x) dx F X ( ) = 0 F X () = F X (x) is a nondecreasing function of x Expectation Operator E{g(X)} =. Mean g(x) = x 2. 2nd moment g(x) = x 2 g(x) f X (x)dx 3. Variance g(x) = (x E{X}) 2 4. Relation between mean, 2nd moment, and variance σ X 2 = E{X 2 } (E{X}) 2 Two Random Variables. Bivariate probability density function f XY (x, y) P{a < X b, c < Y d} = b d a c f XY (x,y)dxdy f XY (x,y)dxdy = f XY (x, y) 0 2. Marginal probability density functions f X (x) and f Y (y) f X (x) = f XY (x, y)dy

6 May EE 438 Essential Definitions... 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} = 5. Covariance of X and Y 2 σ XY xyf XY (x, y)dxdy = E{(X E{X})(Y E{Y})} = E{XY} E{X}E{Y} 6. Correlation coefficient of X and Y ρ XY = σ 2 XY σ X σ Y 0 ρ XY 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. 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]}= µ 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]} 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], then X[n] and Y[n] are jointly widesense stationary, and m µ X = µ Y h[m] m r XY [n] = h[n] r XX [n] r YY [n] = h[n] r XY [ n] 2D CS Signals and Operators, x, y < / 2. rect(x, y) = 0, x, y > / 2 2. sinc(x,y) = sin(πx ) πx sin(πy) πy 3. circ( x,y) =, x2 + y 2 < / 2 0 x 2 + y 2 > / 2 4. jinc(x,y) = J ( π x 2 + y ) 2 2 x 2 + y 2 5. δ(x, y) = lim Δ 0 Δ 2 rect x Δ, y Δ 6. f (x, y) g(x, y) = f (x ξ, y η)g( ξ, η)dξdη

7 May EE 438 Essential Definitions rep XY [ f (x, y)] = k = l= comb XY [ f (x, y)] = k = l= 9. Separability f (x kx,y ly) f (kx,ly )δ(x kx,y ly) f (x, y) = g(x) h(y) Properties of the CS Impulse. 2. δ (x, y) dxdy = and δ(x, y) = 0, (x, y) (0,0) δ (x x 0, y y 0 ) f (x,y)dxdy = f (x 0, y 0 ), provided f (x, y) is continuous at (x,y) = (x 0, y 0 ) δ(x x 0, y y 0 ) f (x,y) = 3. f (x x 0, y y 0 ) CSFT Sufficient conditions for existence or f (x, y) dxdy < F(u, v) = f (x, y) 2 dxdy < f (x, y)e j2π (ux+ vy) dxdy Inverse transform f (x, y) = F(u, v)e j2 π (ux+ vy). Linearity a f (x,y) + a 2 f 2 (x, y) CSFT a F (u, v) + a 2 F 2 (u, v) 2. Scaling and shifting x x f 0, y y 0 CSFT a b dudv ab F(au, bv)e j2π (ux 0 +vy 0 ) f (x, y)e j 2π (u 0 x+ v 0 y) CSFT F(u u 0, v v 0 ) 4. Reciprocity F(x, y) CSFT f ( u, v) 5. Parseval f (x, y) 2 dxdy = F(u, v) 2 dudv 6. Initial Value F(0, 0) = f (x, y) dxdy x(0,0) = F(u, v) dudv 7. Convolution f (x, y) g(x, y) CSFT F(u, v)g(u, v) 8. Product f (x, y)g(x,y) CSFT F(u, v) G(u, v)

8 May EE 438 Essential Definitions Transform of doubly periodic signal rep XY [ f (x, y)] CSFT XY comb XY 0. Transform of sampled signal CSFT comb XY [ f (x, y)] XY rep XY [F(u, v)] [F(u, v)] 2. N roots z k, k = 0,K, N, of a complex constant u 0, i.e. N solutions to z N u 0 = 0 / N = u 0 e j ( 2 πk+ u 0) / N, k = 0,K,N z k. Transform of separable signal g(x) h(y) CSFT G(u) H(v). rect(x, y) CSFT sinc(u,v) 2. circ( x,y) CSFT jinc( u, v) 3. δ(x, y) CSFT and CSFT δ (u, v) 4. e j2π(u 0 x +v 0 y) CSFT δ (u u 0,v v 0 ) 5. cos[ 2π(u 0 x + v 0 y) ] CTFT 2 {δ(u u 0,v v 0 ) + δ(u + u 0,v + v 0 )} Computed Tomography. Radon Transform (Projection) p θ (t) = f (x, y)δ(x cosθ + y sinθ t)dxdy 2. Fourier Slice Theorem P θ (ρ) = p θ (t) e j 2πρ t dt = F(ρ cosθ, ρsinθ) Miscellaneous. Geometric Series N z n n= 0 = z N z