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A STABLE, ONE-MULTIPLY-PER-STEP, ALGORITHM FOR DIGITAL GENERATION OF SINUSOIDS IN REAL TIME

IP.com Disclosure Number: IPCOM000128000D
Original Publication Date: 1976-Dec-31
Included in the Prior Art Database: 2005-Sep-14
Document File: 5 page(s) / 19K

Publishing Venue

Software Patent Institute

Related People

Stefan M. Silverston: AUTHOR [+3]

Abstract

- A new algorithm for the generation of sinusoids in real time is pri*sc*nt.c*cl .:nnd :m :a 1 yne*d . 'fhc* new algorithm Is compared to previously-known algorithms, and shown to he stable and to .require only one multiplication per cycle. A STABLE, ONE-MULTIPLY-PER-STEP, ALGORITHM FOR DIGITAL GENERATION OF SINUSOIDS IN REAL TIME by Stefan M. Silverston

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

A STABLE, ONE-MULTIPLY-PER-STEP, ALGORITHM FOR DIGITAL GENERATION OF SINUSOIDS IN REAL TIME

by

Stefan M. Silverston

Technical Report # 76-9 March, 1976

Abstract

- A new algorithm for the generation of sinusoids in real time is pri*sc*nt.c*cl .:nnd :m :a 1 yne*d
. 'fhc* new algorithm Is compared to previously-known algorithms, and shown to he stable and to .require only one multiplication per cycle.

A STABLE, ONE-MULTIPLY-PER-STEP, ALGORITHM FOR DIGITAL GENERATION OF SINUSOIDS IN REAL TIME by Stefan M. Silverston

Standard Schemes

There are two basic schemes which are applicable to the construction of digital oscillators, that is, to the digital generation of sinusoids in real time. These are:

1. stored-function read-only memory schemes

2. schemes based upon the identities

(Equation Omitted)

It is, in principle, possible to represent sinusoids to any desired accuracy in read-only memory. However, reasons of economy may make the use of read-only memory undesirable in many, practical situations. Schemes based upon the trigonometric identities are cheaper from the standpoint of hardware requirements.

If we let the algorithm

(Equation Omitted)

generates

(Equation Omitted)

(Equation Omitted)

Iowa State University Page 1 Dec 31, 1976

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A STABLE, ONE-MULTIPLY-PER-STEP, ALGORITHM FOR DIGITAL GENERATION OF SINUSOIDS IN REAL TIME

This algorithm is the one commonly used in analog computers It in corporates no divergence from sinusoidality. The amplitude of the sinu- soids generated is a constant based on initial conditions. Hence there is no inherent instability in this system. The only source of error is round-off.

One-Multiply Scheme: the CORDIC Algorithm An additional economy can be realized if the number of multiplica-tions is reduced, for this reduces the number of multiplier chips needed. The algorithm 3 requires two multiplications per step. There is a savings if this is reduced to one multiplication per step. The CORDIC algorithm (2] represents a modification of 3 with only one multiplication per step. The CORDIC scheme uses, in effect, the approx-imation

(Equation Omitted)

to give the algorithm 6.

(Equation Omitted)

We show that, if we define q~ by

the expressions analogous to 4 are

(Equation Omitted)

Note that the expressions $, unlike 4, are unbounded, because of the (Cos ~)n denominators. The CORDIC algorithm produces near-sinusoids. with monotonically increasing amplitudes. Aside from any distortion which results from this effect, the CORDIC algorithm is inherently un- stable: any registers used to hold results of CORDIC computation will ultimately overflow, unless supplementary measures are included in the scheme for the explicit purpose of preventing this.

The expressions.8 may be verified by substituting 7 and 8 into 6

(Equation Omitted)

4

and noticing that the expressions 8 reduce to co and so for n = 0.

Another way of looking at the n-dependent amplitud...