Appendix B

Mathematical Relations

B.1 Trigonometric Identities

cos (a \pm b) = cos (a) cos (b) \mp sin (a) sin (b) (B .1)

$\begin{array}{l} cos (a \pm b) = cos (a) cos (b) \mp sin (a) sin (b) & (B .1) \end{array}$

sin (a \pm b) = sin (a) cos (b) \mp cos (a) sin (b) (B .2)

$\begin{array}{l} sin (a \pm b) = sin (a) cos (b) \mp cos (a) sin (b) & (B .2) \end{array}$

tan (a \pm b) = tan ( a ) \pm tan ( b ) 1 \mp tan ( a ) tan ( b ) (B .3)

$\begin{array}{l} tan (a \pm b) = \frac{tan (a) \pm tan (b)}{1 \mp tan (a) tan (b)} & (B .3) \end{array}$

cos (a) cos (b) = 1 2 cos (a + b) + 1 2 cos (a - b) (B .4)

$\begin{array}{l} cos (a) cos (b) = \frac{1}{2} cos (a + b) + \frac{1}{2} cos (a - b) & (B .4) \end{array}$

sin (a) sin (b) = 1 2 cos (a - b) - 1 2 cos (a + b) (B .5)

$\begin{array}{l} sin (a) sin (b) = \frac{1}{2} cos (a - b) - \frac{1}{2} cos (a + b) & (B .5) \end{array}$

sin (a) cos (b) = 1 2 sin (a + b) + 1 2 sin (a - b) (B .6)

$\begin{array}{l} sin (a) cos (b) = \frac{1}{2} sin (a + b) + \frac{1}{2} sin (a - b) & (B .6) \end{array}$

cos (a) + cos (b) = 2 cos (a + b 2) cos (a - b 2) (B .7)

$\begin{array}{l} cos (a) + cos (b) = 2 cos (\frac{a + b}{2}) cos (\frac{a - b}{2}) & (B .7) \end{array}$

cos (a) - cos (b) = - 2 s i n (a + b 2) sin (a - b 2) (B .8)

$\begin{array}{l} cos (a) - cos (b) = - 2 s i n (\frac{a + b}{2}) sin (\frac{a - b}{2}) & (B .8) \end{array}$

sin (a) + sin (b) = 2 s i n (a + b 2) cos (a - b 2) (B .9)

$\begin{array}{l} sin (a) + sin (b) = 2 s i n (\frac{a + b}{2}) cos (\frac{a - b}{2}) & (B .9) \end{array}$

sin (a) - sin (b) = 2 s i n (a - b 2) cos (a + b 2) (B .10)

$\begin{array}{l} sin (a) - sin (b) = 2 s i n (\frac{a - b}{2}) cos (\frac{a + b}{2}) & (B .10) \end{array}$

cos (2 a) = cos 2 (a) - sin 2 (a) (B .11)

$\begin{array}{l} cos (2 a) = {cos}^{2} (a) - {sin}^{2} (a) & (B .11) \end{array}$

sin (2 a) = 2 sin (a) cos (a) (B .12)

$\begin{array}{l} sin (2 a) = 2 sin (a) cos (a) & (B .12) \end{array}$

tan (2 a) = 2 tan ( a ) 1 - tan 2 ( a ) (B .13)

$\begin{array}{l} tan (2 a) = \frac{2 tan (a)}{1 - {tan}^{2} (a)} & (B .13) \end{array}$

cos 2 (a) = 1 2 + 1 2 cos (2 a) (B .14)

$\begin{array}{l} {cos}^{2} (a) = \frac{1}{2} + \frac{1}{2} cos (2 a) & (B .14) \end{array}$

sin 2 (a) = 1 2 - 1 2 cos (2 a) (B .15)

$\begin{array}{l} {sin}^{2} (a) = \frac{1}{2} - \frac{1}{2} cos (2 a) & (B .15) \end{array}$

\int x e a x d x = a x - 1 a 2 e a x (B .16)

$\begin{array}{l} \int x e^{a x} d x = \frac{a x - 1}{a^{2}} e^{a x} & (B .16) \end{array}$

\int 1 x 2 + a 2 d x = 1 a tan - 1 (x a) (B .17)

$\begin{array}{l} \int \frac{1}{x^{2} + a^{2}} d x = \frac{1}{a} {tan}^{- 1} (\frac{x}{a}) & (B .17) \end{array}$

\int x x 2 + a 2 d x = 1 2 ln (x 2 + a 2) (B .18)

$\begin{matrix} \int \frac{x}{x^{2} + a^{2}} d x = \frac{1}{2} ln (x^{2} + a^{2}) & (B .18) \end{matrix}$

\int x 2 x 2 + a 2 d x = x - a tan - 1 (x a) (B .19)

$\begin{matrix} \int \frac{x^{2}}{x^{2} + a^{2}} d x = x - a {tan}^{- 1} (\frac{x}{a}) & (B .19) \end{matrix}$

\int x cos (a x) d x = 1 a 2 [cos (a x) + a x sin (a x)] (B .20)

$\begin{matrix} \int x cos (a x) d x = \frac{1}{a^{2}} [cos (a x) + a x sin (a x)] & (B .20) \end{matrix}$

\int x sin (a x) = 1 a 2 [sin (a x) - a x cos (a x)] (B .21)

$\begin{matrix} \int x sin (a x) = \frac{1}{a^{2}} [sin (a x) - a x cos (a x)] & (B .21) \end{matrix}$

\int e a x cos (b x) d x = e a x a 2 + b 2 [a cos (b x) + b sin (b x)] (B .22)

$\begin{matrix} \int e^{a x} cos (b x) d x = \frac{e^{a x}}{a^{2} + b^{2}} [a cos (b x) + b sin (b x)] & (B .22) \end{matrix}$

\int e a x sin (b x) d x = e a x a 2 + b 2 [a sin (b x) - b cos (b x)] (B .23)

$\begin{matrix} \int e^{a x} sin (b x) d x = \frac{e^{a x}}{a^{2} + b^{2}} [a sin (b x) - b cos (b x)] & (B .23) \end{matrix}$

\int (a + b x) n d x = ( a + b x ) n + 1 b ( n + 1 ), n > 0 (B .24)

$\begin{matrix} \int {(a + b x)}^{n} d x = \frac{{(a + b x)}^{n + 1}}{b (n + 1)}, & n > 0 & (B .24) \end{matrix}$

\int 1 ( a + b x ) n d x = - 1 b ( n - 1 ) ( a + b x ) n - 1, n > 1 (B .25)

$\begin{matrix} \int \frac{1}{{(a + b x)}^{n}} d x = \frac{- 1}{b (n - 1) {(a + b x)}^{n - 1}}, & n > 1 & (B .25) \end{matrix}$

Signal	Transform	ROC
δ[n]	1	all z
u[n]	$z z - 1$ $\frac{z}{z - 1}$	\|z\| > 1
u[−n]	$- z z - 1$ $\frac{- z}{z - 1}$	\|z\| < 1
an u[n]	$z z - a$ $\frac{z}{z - a}$	\|z\| > \|a\|
−an u[−n − 1]	$z z - a$ $\frac{z}{z - a}$	\|z\| < \|a\|
cos (Ω0n) u[n]	z[z−cos(Ω0)]z2−2cos(Ω0) z+1 $\frac{z [z - cos (Ω_{0})]}{z^{2} - 2 cos (Ω_{0}) z + 1}$	\|z\| > 1
sin (Ω0n) u[n]	$sin ( Ω 0 ) z z 2 - 2 cos ( Ω 0 ) z + 1$ $\frac{sin (Ω_{0}) z}{z^{2} - 2 cos (Ω_{0}) z + 1}$	\|z\| > 1
an cos (Ω0n) u[n]	$z [ z - a cos ( Ω 0 ) ] z 2 - 2 a cos ( Ω 0 ) z + a 2$ $\frac{z [z - a cos (Ω_{0})]}{z^{2} - 2 a cos (Ω_{0}) z + a^{2}}$	\|z\| > \|a\|
an sin (Ω0n) u[n]	$a sin ( Ω 0 ) z x 2 - 2 a cos ( Ω 0 ) z + a 2$ $\frac{a sin (Ω_{0}) z}{x^{2} - 2 a cos (Ω_{0}) z + a^{2}}$	\|z\| > \|a\|
nan u[n]	$a z ( z - a ) 2$ $\frac{a z}{{(z - a)}^{2}}$	\|z\| > \|a\|

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