By Alexander D. Poularikas
This ebook includes numerous topics for sign processing and, first-class selection of SP information, and has been a superb aid for me to memind the very mathematical expression and formulation which i can't determine on my head. so much extraordinary issue to me used to be the prepared checklist of formulation which you have by no means present in any SP e-book. This publication will be used for fast reference for sign processing know-how, and while you're within the entrance a part of sign processing expertise, you could have to use the various formulation, so, it could be a superb aid in your job equivalent to writing a technical paper, discovering a formal use of formulation, realizing quick on a mathematics-related matters and so forth. it'd be one in all crucial books that you should locate in your bookshelf for a very long time.
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Extra resources for Handbook of Formulas and Tables for Signal Processing (Electrical Engineering Handbook)
Since (s − jnπ) e st 0 lim = , r→0 s (1 + e − s ) 0 s→ jnπ this expression is evaluated and yields e jnπt/jnπ. Thus, for all poles, I3 = 1 2 πj ∫ π/2 −π/2 r→0 s→ jnπ e st ds s (1 + e − s ) ∞ ∞ sin nπt jπ e jnπt 1 1 1 2 = + = + 2 πj n=−∞ jnπ 2 2 2 π n=1 n n odd n odd ∑ ∑ Finally, the residues enclosed within the contour are ∞ Res e st e jnπt 1 2 1 = + = + −s s (1 + e ) 2 n=−∞ jnπ 2 π ∑ n odd ∞ ∑ n =1 n odd sin nπt n which is seen to be twice the value around the hooks.
1 2π Inverse transform ∞ ∫ F (ω ) e jω t dω F(ω ) −∞ 3. Linearity af1(t) + bf2(t) aF1(ω) + bF2(ω) 4. Symmetry F (t ) 2π f (–ω) 5. Time shifting f (t ± t o ) e ± jω to F(ω ) 6. Scaling f ( at ) 1 ω F a a 7. Frequency shifting 8. Modulation 9. Time differentiation F (ω m ω o ) e ± jω ot f (t ) f (t ) cos ω o t f (t )sin ω o t [ F(ω + ω o ) + F(ω − ω o )] [ F(ω − ω o ) − F(ω + ω o )] 1 2 1 2j dn f (t ) dt n ( jω ) n F(ω ) ∞ 10. Time convolution ∫ f (t ) ∗ h(t ) = f ( τ) h(t − τ) dτ F (ω ) H (ω ) −∞ 11.
Dt n [Ln(t) is the Laguerre polynomial of degree n] 52 1 ( s + a) n t ( n −1) e − at (n − 1)! 3 Table of Laplace Transforms (continued) F(s) 125 s 3 + a2 s 2 + a1 s + ao (s + b)(s + c)(s + d )(s + f ) 2 126 s ( s − a) 3 / 2 127 s−a − s−b 128 1 s +a f(t) − b 3 + a2 b 2 − a1 b + ao − bt −c 3 + a2 c 2 − a1c + ao − ct e + e 2 (b − c)(d − c)( f − c) 2 (c − b)(d − b)( f − b) − d 3 + a2 d 2 − a1 d + ao − dt − f 3 + a2 f 2 − a1 f + ao − ft te e + + (b − d )(c − d )( f − d ) 2 (b − f )(c − f )(d − f ) ( − f 3 + a2 f 2 − a1 f + ao )[(b − f )(c − f ) 2 + 3 f − 2 a2 f + a1 e − ft − + (b − f )(d − f ) + (c − f )( d − f )] e − ft ( b − f ) 2 (c − f ) 2 ( d − f ) 2 (b − f )(c − f )(d − f ) 1 at e (1 + 2 at ) πt 1 2 πt3 (e bt − e at ) 2 1 − ae a t erfc ( a t ) πt 129 s s − a2 2 1 + ae a t erf ( a t ) πt 130 s s + a2 2 1 − 2a e − a t πt π 131 1 s (s − a 2 ) 1 a 2t e erf ( a t ) a 132 1 s (s + a 2 ) 2 2 e −a t a π 133 b2 − a2 (s − a 2 )(b + s ) 134 1 s ( s + a) 135 1 ( s + a) s + b 136 b2 − a2 s (s − a 2 )( s + b) ∫ a t ∫ a t e λ dλ 2 0 e λ dπ 2 0 2 2 e a t [b − a erf ( a t )] − be b t erfc (b t ) 2 e a t erfc ( a t ) 1 e − at erf ( b − a t ) b−a 2 b 2 e a t erf ( a t ) − 1 + e b t erfc (b t ) a 137 (1 − s) s n +(1 / 2 ) n!