Goertzel Algorithm - Digital Signal Processing and Applications with the C6713 and C6416 DSK - page 494

which can be written as

N -

 0

1

() = ()

( )

xrW N kN r

--

Xk (G.2)

r =

Define a discrete-time function as

N -

 0

1

( )

() = () --

kn r

yn xrW (G.3)

k N

r =

The discrete transform is then

() = () = Xk y n

k

(G.4)

n N

Equation (G.3) is a discrete convolution of a finite-duration input sequence x ( n ),
0

1, with the infinite sequence W - kn . The infinite impulse response is < n < N -

therefore

() = W N kn

-

hn (G.5)

The Z-transform of h ( n ) in (G.5) is

•
 0

() = () -

Hz hnz n (G.6)

n =

Substituting (G.5) into (G.6) gives

•

1

Â

() = Hz W

- kn

z

- n

= + 1 W z

- k - 1

+ W

- 2 k

z

- 2

+◊◊◊ = (G.7)

N N N

-- 2 k 1

1 - Wz

n = 0 N

Thus equation (G.7) represents the transfer function of the convolution sum in
equation (G.3). Its flow graph represents the first-order Goertzel algorithm and is
shown in Figure G.1. The DFT of the k th frequency component is calculated by

x ( n ) y ( n )

+
+

Z -1

W N - K
FIGURE G.1. First-order Goertzel algorithm.