Tables in Signals and Systems [PDF]

Tables in Signals and Systems Magnus Lundberg1 Revised October 1999

Contents I

Continuous-time Fourier series I-A Properties of Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . I-B Fourier series table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 3

II Continuous-time Fourier transform II-A Properties of the Fourier transform . . . . . . . . . . . . . . . . . . . . . . . II-B Fourier transform table . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 4 5

III Discrete-time Fourier series III-AProperties of discrete-time Fourier series . . . . . . . . . . . . . . . . . . . . III-B Fourier series table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8

IV Discrete-time Fourier transform IV-A Properties of the discrete-time Fourier transform . . . . . . . . . . . . . . . IV-B Discrete-time Fourier transform table . . . . . . . . . . . . . . . . . . . . . .

9 9 10

V Sampling and reconstruction

11

VI Z-transform VI-A Properties of the Z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . VI-B Z-transform table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 12 13

1

The major part of this collection of tables was originally developed at the Div. of Signal Processing,

Lule˚ a University of Technology. It has been revised by Magnus Lundberg in October 1999

TABLES IN SIGNALS AND SYSTEMS, OCT. 1999

2

Definitions 4

sinc (t) =

sin (πt) πt

4

Ωo =

2π T0

I. Continuous-time Fourier series A. Properties of Fourier series Periodic signal ∞ X

x (t) =

Fourier serie coefficient Z 4 1 x (t) e−jkΩo t dt ak = To To

ak ejkΩo t

k=−∞

x(t) y(t)



ak bk

Periodic with periodT0

Ax (t) + By (t)

Aak + Bbk

x (t − t0 )

ak e−jk(2π/T0 )t0

ejM (2π/T0 )t x (t)

ak−M

x∗ (t)

a∗−k

x (−t)

a−k

x (αt) , α > 0 (Periodic with period T0 /α) Z x (τ ) y (t − τ ) dτ

ak T0 ak bk

T0 ∞ X

x (t) y (t)

al bk−l

l=−∞

d x (t) dt Z t x (τ ) dτ (Bounded and periodic only if a0 = 0) −∞

jk

2π ak T0

1 ak jk (2π/T0 )

If x (t) is real valued then  ak = a∗−k    < {ak } = < {a−k } = {ak } = −= {a−k }    |ak | = |a−k | arg {ak } = − arg {a−k }

x (t)

xe (t) = E{x(t)} xo (t) = O{x(t)}