Table 1: Properties of the Continuous-Time Fourier Series. Property Periodic Signal Fourier Series Coefficients

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1 able : Properties of the Continuous-ime Fourier Series x(t = e jkω0t = = x(te jkω0t dt = e jk(/t x(te jk(/t dt Property Periodic Signal Fourier Series Coefficients x(t y(t } Periodic with period and fundamental frequency ω 0 =/ b k Linearity Ax(t+By(t A + Bb k ime-shifting x(t t 0 e jkω 0t 0 = e jk(/t 0 Frequency-Shifting e jmω0t = e jm(/t x(t M Conjugation x (t ime Reversal x( t a k a k ime Scaling x(αt,α>0 (periodic with period /α Periodic Convolution x(τy(t τdτ b k Multiplication Differentiation Integration Conjugate Symmetry for Real Real and Even x(ty(t dx(t dt t x(tdt x(t real x(t real and even (finite-valued and periodic only if a 0 =0 l= a l b k l jkω 0 = jk ( ( = jk(/ jkω 0 = a k Re } = Rea k } Im } = Ima k } = a k < = < a k real and even Real and Odd x(t realandodd purely imaginary and odd Even-Odd Decomposition of Real xe (t =Evx(t} [x(t real] x o (t =Odx(t} [x(t real] Parseval s Relation for Periodic x(t 2 dt = 2 Re } jim }

2 able 2: Properties of the Discrete-ime Fourier Series x[n] = e jkω0n = e jk(/n = k=<> k=<> x[n]e jkω0n = n=<> n=<> x[n]e jk(/n Property Periodic signal Fourier series coefficients x[n] y[n] } Periodic with period and fundamental frequency ω 0 =/ b k } Periodic with period Linearity Ax[n]+By[n] A + Bb k ime shift x[n n 0 ] e jk(/n 0 Frequency Shift e jm(/n x[n] M Conjugation x [n] ime Reversal x[ n] a k a k ime Scaling x (m [n] = Periodic Convolution Multiplication x[n/m] if n is a multiple of m 0 if n is not a multiple of m (periodic with period m x[r]y[n r] r= x[n]y[n] m b k l= ( viewed as periodic with period m a l b k l First Difference x[n] x[n ] ( e jk(/ n ( ( finite-valued and Running Sum x[k] a periodic only if a 0 =0 ( e jk(/ k = a k Re } = Rea k } Conjugate Symmetry x[n] real Ima for Real k } = Ima k } = a k < = < a k Real and Even x[n] real and even real and even Real and Odd x[n] realandodd purely imaginary and odd Even-Odd Decomposition of Real x e [n] =Evx[n]} x o [n] =Odx[n]} [x[n] real] [x[n] real] Re } jim } Parseval s Relation for Periodic x[n] 2 = 2 n= k=

3 able 3: Properties of the Continuous-ime Fourier ransform x(t = X(jωe jωt dω X(jω= x(te jωt dt Property Aperiodic Signal Fourier transform x(t y(t X(jω Y (jω Linearity ax(t+by(t ax(jω+by (jω ime-shifting x(t t 0 e jωt 0 X(jω Frequency-shifting e jω0t x(t X(j(ω ω 0 Conjugation x (t X ( jω ime-reversal x( t X( jω ( jω ime- and Frequency-Scaling x(at a X a Convolution x(t y(t X(jωY (jω Multiplication x(ty(t X(jω Y (jω d Differentiation in ime dt x(t jωx(jω t Integration x(tdt jω X(jω+X(0δ(ω Differentiation in Frequency tx(t j d dω X(jω Conjugate Symmetry for Real Symmetry for Real and Even Symmetry for Real and Odd Even-Odd Decomposition for Real x(t real x(t real and even x(t real and odd x e (t =Evx(t} x o (t =Odx(t} [x(t real] [x(t real] X(jω=X ( jω ReX(jω} = ReX( jω} ImX(jω} = ImX( jω} X(jω = X( jω < X(jω = < X( jω X(jω real and even X(jω purely imaginary and odd ReX(jω} jimx(jω} Parseval s Relation for Aperiodic + x(t 2 dt = + X(jω 2 dω

4 able 4: Basic Continuous-ime Fourier ransform Pairs Fourier series coefficients Signal Fourier transform (if periodic e jω 0t e jkω 0t δ(ω ω 0 δ(ω kω 0 cos ω 0 t [δ(ω ω 0 +δ(ω + ω 0 ] sin ω 0 t x(t = Periodic square wave, t < x(t = 0, < t 2 and x(t + =x(t j [δ(ω ω 0 δ(ω + ω 0 ] δ(ω 2sinkω 0 δ(ω kω 0 k ( δ ω k a = =0, otherwise a = a = 2 =0, otherwise a = a = 2j =0, otherwise a 0 =, =0,k 0 ( this is the Fourier series representation for any choice of >0 ω 0 δ(t n = n=, t < 2sinω x(t 0, t > ω sin Wt, ω <W X(jω= t 0, ω >W δ(t u(t jω + δ(ω δ(t t 0 e jωt 0 e at u(t, Rea} > 0 te at u(t, Rea} > 0 t n (n! e at u(t, Rea} > 0 a + jω (a + jω 2 (a + jω n sinc ( kω0 for all k = sin kω 0 k

5 able 5: Properties of the Discrete-ime Fourier ransform x[n] = X(e jω e jωn dω X(e jω = n= x[n]e jωn Property Aperiodic Signal Fourier transform x[n] } X(e jω Periodic with y[n] Y (e jω period Linearity ax[n]+by[n] ax(e jω +by (e jω ime-shifting x[n n 0 ] e jωn 0 X(e jω Frequency-Shifting e jω0n x[n] X(e j(ω ω0 Conjugation x [n] X (e jω ime Reversal x[ n] X(e jω x[n/k], if n = multiple of k ime Expansions x (k [n] = X(e jkω 0, if n multiple of k Convolution x[n] y[n] X(e jω Y (e jω Multiplication x[n]y[n] X(e jθ Y (e j(ω θ dθ Differencing in ime x[n] x[n ] ( e jω X(e jω n Accumulation x[k] e jω X(ejω +X(e j0 δ(ω k Differentiation in Frequency nx[n] j dx(ejω dω X(e jω =X (e jω ReX(e jω } = ReX(e jω } Conjugate Symmetry for x[n] real ImX(e jω } = ImX(e jω } Real X(e jω = X(e jω < X(e jω = < X(e jω Symmetry for Real, Even Symmetry for Real, Odd Even-odd Decomposition of Real x[n] real and even x[n] realandodd x e [n] =Evx[n]} x o [n] =Odx[n]} [x[n] real] [x[n] real] X(e jω real and even X(e jω purely imaginary and odd ReX(e jω } jimx(e jω } Parseval s Relation for Aperiodic x[n] 2 = X(e jω 2 dω n=

6 able 6: Basic Discrete-ime Fourier ransform Pairs Fourier series coefficients Signal Fourier transform (if periodic k= e jω 0n cos ω 0 n sin ω 0 n x[n] = e jk(/n Periodic square wave, n x[n] = 0, < n /2 and x[n + ] =x[n] δ[n k] a n u[n], a < x[n] j l= l= l= l=, n sin[ω( + 2 ] 0, n > sin(ω/2 sin Wn n = W sinc ( Wn 0 <W < ( δ ω k (a ω 0 = m, k = m, m ±,m ± 2,... δ(ω ω 0 l = 0, otherwise ω (b 0 irrational he signal is aperiodic (a ω 0 = m δ(ω ω 0 l+δ(ω + ω 0 l} = 2, k = ±m, ±m ±,±m ± 2,... 0, otherwise ω (b 0 irrational he signal is aperiodic (a ω 0 = r 2j, k = r, r ±,r ± 2,... δ(ω ω 0 l δ(ω + ω 0 l} = 2j, k = r, r ±, r ± 2,... 0, otherwise ω (b 0 irrational he signal is aperiodic, k =0, ±,±2,... δ(ω l = 0, otherwise ( δ ω k ( δ ω k = ae jω, 0 ω W X(ω = 0, W < ω X(ωperiodic with period δ[n] u[n] + e jω + δ(ω k δ[n n 0 ] e jωn 0 (n +a n u[n], a < ( ae jω 2 (n + r! a n u[n], a < n!(r! ( ae jω r = sin[(k/(+ 2 ] sin[k/2], k 0, ±,±2,... = 2+, k =0, ±,±2,... for all k

7 able 7: Properties of the Laplace ransform Property Signal ransform ROC x(t X(s R x (t X (s R x 2 (t X 2 (s R 2 Linearity ax (t+bx 2 (t ax (s+bx 2 (s At least R R 2 ime shifting x(t t 0 e st 0 X(s R Shifting in the s-domain e s0t x(t X(s s 0 Shifted version of R [i.e., s is in the ROC if (s s 0 isin R] ime scaling x(at ( s a X a Conjugation x (t X (s R Scaled ROC (i.e., s is in the ROC if (s/a is in R Convolution x (t x 2 (t X (sx 2 (s At least R R 2 Differentiation in the ime Domain Differentiation in the s-domain Integration in the ime Domain t d x(t dt sx(s At least R tx(t d ds X(s R x(τd(τ X(s s At least R Res} > 0} Initial- and Final Value heorems If x(t = 0 for t<0andx(t contains no impulses or higher-order singularities at t =0,then x(0 + = lim s sx(s If x(t = 0 for t<0andx(t has a finite limit as t,then lim t x(t = lim s 0 sx(s

8 able 8: Laplace ransforms of Elementary Functions Signal ransform ROC. δ(t All s 2. u(t s Res} > 0 3. u( t s Res} < 0 4. t n (n! u(t s n Res} > 0 5. tn (n! u( t s n Res} < 0 6. e αt u(t 7. e αt u( t 8. t n (n! e αt u(t 9. tn (n! e αt u( t s + α Res} > α s + α Res} < α (s + α n Res} > α (s + α n Res} < α 0. δ(t e s All s. [cos ω 0 t]u(t 2. [sin ω 0 t]u(t 3. [e αt cos ω 0 t]u(t s s 2 + ω 2 0 ω 0 s 2 + ω0 2 s + α (s + α 2 + ω 2 0 ω 0 Res} > 0 Res} > 0 Res} > α 4. [e αt sin ω 0 t]u(t (s + α 2 + ω0 2 Res} > α 5. u n (t = dn δ(t dt n s n All s 6. u n (t =u(t u(t }} n times s n Res} > 0