Complementary Sequences - Properties of Complementary Pairs of Sequences

Properties of Complementary Pairs of Sequences

• Complementary sequences have complementary spectra. As the autocorrelation function and the power spectra form a Fourier pair complementary sequences also have complementary spectra. But as the Fourier transform of a delta function is a constant we can write

where C_S is a constant.

S_a and S_b are defined as a squared magnitude of the Fourier transform of the sequences. The Fourier transform can be a direct DFT of the sequences, it can be a DFT of zero padded sequences or it can be a continuous Fourier transform of the sequences which is equivalent to the Z transform for Z = ejω.

• CS spectra is upper bounded. As S_a and S_b are non-negative values we can write

also

• If any of the sequences of the CS pair is inverted (multiplied by −1) they remain complementary. More generally if any of the sequences is multiplied by ejφ they remain complementary;
• If any of the sequences is reverted (inverted in time) they remain complementary;
• If any of the sequences is delayed they remain complementary;
• If the sequences are interchanged they remain complementary;
• If both sequences are multiplied by the same constant (real or complex) they remain complementary;
• If both sequences are decimated in time by K they remain complementary. More precisely if from a complementary pair ((a(k), b(k)) we form a new pair (a(Nk), b(N*k)) with zero samples in between then the new sequences are complementary.
• If alternating bits of both sequences are inverted they remain complementary. In general for arbitrary complex sequences if both sequences are multiplied by ejπkn/N (where k is a constant and n is the time index) they remain complementary
• A new pair of complementary sequences can be formed as and where denotes concatenation and a and b are a pair of CS;
• A new pair of sequences can be formed as {a b} and {a −b} where {..} denotes interleaving of sequences.
• A new pair of sequences can be formed as a + b and a − b.