Digital Signal Processing - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 28

The Dirichlet function (or Dirichlet's kernel, or periodic sinc function) diric N ,
parameterized by the integer N ≥ 1 , is defined for all t by:

( )

⎧

sin π tN

if tZ ∉

⎪
= ⎨ ⎪ −

( )

N s in π T

()

diric t [2.17]

N

() (

)

tN − 1

1 if tZ ∈

⎩

The value at the integer abscissa is obtained by continuity extension; it is always
equal to 1 if N is odd, (-1)* if N is even. The Dirichlet's function is even and
periodic with period 1 if N is odd; it is even, periodic with period 2 and symmetrical

1 ,0
2

⎛ ⎜ ⎝ ⎠ if N is even (Figure 2.3). Thus, in all cases, its
absolute value is even and periodic with period 1. It is zero for all non-integer t

in relation to the point

1
N

multiple of . The arch of this function centered on 0 is called the main lobe, the

others are secondary lobes. The main lobe gets narrower and the secondary lobes'
amplitude gets smaller as N increases.

Figure 2.3. Dirichlet functions diric 15 and diric 16

The unit series 1 Z is the series equal to 1; for all k :

()

k =

[2.18]

1 1

Z

The discrete time cisoid of amplitude a , frequency v and initial phase φ is defined

by:

( )

( )

j 2 + π v k φ

[2.19]

ae

kZ ∈

We immediately notice that the frequency of the cisoid is defined modulo 1.