Running discrete cosine transform H. Olkkonen Department Finland

of Applied

Physics,

University

of Kuopio,

PO Box 1627, SF-70211

Kuopio,

Received February 1991, accepted March 1992

ABSTRACT i??udiscretecosinetransform (DCT) has becomean important toolin digital signalprocessingbecauseitspe$ownance is close to the optimal Karhunen-Loeve transform. In this work the running discrete cosine transform (KDCT) is introduced. Using the properties of the discrete Fourier transform kernel W = exp (- 2rr j AV), a fmt recursive algorithm was developedfor real-time computationof the RDCTcoeficients. For N-point KDCT thepresent algorithm needs only 2N real multiplications. The hardware implementationsof the RDCT algorithm and applications in realtime data processing are discussed. Keywords:

Running discrete cosine transform, Fourier transform, fast algorithm

and the inverse transform

INTRODUCTION The discrete cosine transform (DCT)’ has recently become an important tool in signal and image processing applications since it has the most advanced performance among the known fastly computable transforms. The roperties of the DCT are very close to the statistica Ply optimal KarhunenLoeve transform (KLT) for a large number of signal families2333 4. The fast algorithms for computation of the DCT are based on the fast Fourier transform iFFT)4,5 or the direct factorization of the DCT matrix . Recently also the discrete Hartley transform for real-value data has been involved in the computation of the DCT7. Real-time signal processing implementations of the DCT have given rise to an urgent need for the computer inbuilt DCT processors, which are usually based on the concurrent architectures~“. Also the FFT-DCT chip, which possesses the butterfly structure, has been proposed”. In this work we introduce the running discrete cosine transform (RDCT), which is an analogue to the well-known running Fourier transform” employed in the frequency analyser constructions. Using the 2 transform and the properties of the discrete Fourier transform kernel a fast recursive algorithm for realtime computation of the RDCT is developed.

(IDCT) as

N-l

x, = 2

up DCT,

p=o

cos[rp

(2 n + 1)/2N]

where the normalization ap =

VV2

constant

for p = 0

11

(3)

elsewhere

For real-time computation we define discrete cosine transform (RDCT) as RDCT;:

= %

z0

For the real-valued data sequence X, (n= 0, . . ., N- 1) the discrete cosine transform (DCT) is defined as DCTp = %! ? N ” x,co+-p(2n+ ” 0 0 1992 ButterworthHeinemann

the running

&_k cos [np (2k + 1)/2 N]

(4

where n denotes the discrete time. The 2 transform of the sequence x, is defined as 2(x,}

=x(z)=

i X,P *=-li

The Z transform z{xn-k)

=

(5)

of the sequence x,-k

i

2

x,-k is then

-n = z-kx (z)

(6)

?I=-5

The Z transform from

of the RDCT

x,-k cos (@‘(2

RUNNING DISCRETE COSINE TRANSFORM

ap is given by

After rearranging Z(RDCT;}

coefficients

k+ l)/aN)

and using equation

=x(z)

2:

cos(qr(2k+ k

comes

(7)

z-n

(6) we obtain 1)/2N)~-~

0 (8)

1)/2N

(1)

In order to obtain the recursive implementation we replace the transform kernel cos (.) b the real part of the discrete Fourier transform kerne ?

for BES

0141~5425/92/06507-02 J. Biomed. Eng. 19W, Vol. 14, November

507

Running disctete cosine transform: H. Olkkonen

W,=

exp (-2~j/N)

= cos (2r/N) -jsin

(2&N)

= C, - j SN

(9)

Then, the following is valid

Thedatapointsx,(n=n,n-l,n-2,...)maybe reconstructed from the computed RDCT coefficients using the inverse cosine transform (cf. equation [2]) N-l x, = C

p=o

In the above equation the summation as

cp

1N

1 - (- 1)Q-N = (Cp,, 1-

can be written

c-{,z-‘)(l- (-

2-r + z-2

1 - 2 cos (Q/N)

C&Y*

We may note that &., = Ci$ = cos(@/2N). rearranging and using equation (10) we find Z{RDCT;}

l)VN) By

= 1 -