On codes with spectral nulls at rational submultiples of the symbol frequency

Abstract

In digital data transmission (respectively, storage systems), line codes (respectively, recording codes) are used to tailor the spectrum of the encoded sequences to satisfy constraints imposed by the channel transfer characteristics or other system requirements. For instance, pilot tone insertion requires codes with zero mean and zero spectral density at tone frequencies. Embedded tracking/focus servo signals produce similar needs. Codes are studied with spectral nulls at frequencies <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f=kf_{s}/n</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</tex> , is the symbol frequency and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k, n</tex> are relatively prime integers with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k \leq n;</tex> in other words, nulls at rational submultiples of the symbol frequency. A necessary and sufficient condition is given for a null at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</tex> in the form of a finite discrete Fourier transform (DFT) running sum condition. A corollary of the result is the algebraic characterization of spectral nulls which can be simultaneously realized. Specializing to binary sequences, we describe canonical Mealy-type state diagrams (directed graphs with edges labeled by binary symbols) for each set of realizable spectral nulls. Using the canonical diagrams, we obtain a frequency domain characterization of the spectral null systems obtained by the technique of time domain interleaving.