Digital Communications

Constant-Amplitude Zero-Autocorrelation

Q: How is a Zadoff-Chu sequence constructed and why is it CAZAC?

A Zadoff-Chu sequence of length N is defined as x[n] = exp(-j*pi*u*n*(n+1)/N) for odd N, where u is the root index, coprime to N. Every sample lies on the unit circle, so the magnitude is constant at one, satisfying the constant-amplitude property. Because the phase advances quadratically with n, the periodic autocorrelation collapses to zero at every nonzero cyclic shift, giving the zero-autocorrelation property. When N is prime, the magnitude of the cross-correlation between two distinct roots is exactly sqrt(N) (i.e. 1/sqrt(N) of the autocorrelation peak), which is why LTE assigns different roots to neighboring cells. These two properties together are what define a CAZAC sequence, and Zadoff-Chu is the most widely deployed family.

Q: Why do CAZAC sequences have a peak-to-average power ratio of 0 dB?

In the time domain every sample of a CAZAC sequence has identical magnitude, so the instantaneous power never deviates from the average and the PAPR is exactly 0 dB. A constant envelope lets a power amplifier run near saturation without the back-off that high-PAPR OFDM payloads require, which improves efficiency and coverage. This is the main reason 3GPP chose a Zadoff-Chu CAZAC sequence for the LTE PRACH preamble and the uplink sounding and demodulation reference signals: the user equipment can transmit reference symbols at full power. Note the flat envelope holds for the discrete sequence; after pulse shaping or DFT precoding the continuous waveform shows a small but bounded PAPR increase.

Q: How does the ideal autocorrelation improve timing and channel estimation?

Because the periodic autocorrelation is a single impulse at zero shift and zero elsewhere, a correlator at the receiver produces a sharp, unambiguous peak when the local replica aligns with the received sequence. This gives sample-accurate timing for frame and symbol synchronization, refinable to sub-sample precision by interpolating around the peak, and lets the receiver resolve individual multipath taps without sidelobe leakage masking weak echoes. In the frequency domain the constant magnitude means every subcarrier is excited equally, so dividing the received spectrum by the known sequence yields a well-conditioned least-squares channel estimate with uniform noise across the band. The LTE primary synchronization signal exploits exactly this behavior for initial cell search.

/KAY-zak/ (CAZAC)

Often abbreviated CAZAC, this is a family of complex-valued digital sequences engineered so that every time-domain sample has the same magnitude while the periodic autocorrelation equals zero at all nonzero shifts, leaving a single impulse at zero lag. The flat envelope yields a 0 dB peak-to-average power ratio that lets a transmitter run near saturation, and the impulse-like correlation produces a sharp, unambiguous timing peak at the receiver. These twin properties make CAZAC sequences, most notably the Zadoff-Chu sequence family, the workhorse for cell-search synchronization, random-access preambles, channel sounding, and pilot reference signals in LTE, 5G NR, and other OFDM systems.

Category: Digital Communications

PAPR: 0 dB (constant envelope)

Periodic autocorr.: Impulse at zero lag

Why Flat Envelope and Impulse Correlation Matter

CAZAC is not a single sequence but a defining pair of properties that several mathematical families satisfy. The first property, constant amplitude, means all N samples sit on a circle of identical radius in the complex plane, differing only in phase. The second, zero periodic autocorrelation, means a circular correlation of the sequence with a shifted copy of itself returns zero energy unless the shift is zero. A waveform with these characteristics behaves almost ideally for the two jobs a synchronization signal must perform: it transmits efficiently because the amplifier sees no envelope variation, and it locates itself precisely in time because the correlator output has no sidelobes to confuse the detector.

The most important construction is the Zadoff-Chu sequence, whose quadratic phase progression simultaneously guarantees both properties for any length N when the root index u is coprime to N. When N is prime the cross-correlation between two distinct roots has constant magnitude √N, that is 1/√N of the autocorrelation peak, which lets cellular networks assign distinct roots to neighboring cells so receivers can separate overlapping preambles. Other CAZAC families exist, including Frank-Zadoff and Chu polyphase sequences, but Zadoff-Chu dominates because 3GPP standardized it for the LTE and 5G NR primary synchronization signal, the physical random access channel preamble, and the uplink demodulation and sounding reference signals.

Engineers care about CAZAC properties because they decouple two design goals that usually conflict. High-order modulation and OFDM payloads produce large envelope swings that force power-amplifier back-off; a constant-envelope reference signal lets the same hardware push synchronization and pilot symbols at full power, extending uplink range. At the same time the impulse autocorrelation gives the receiver fine timing resolution and a well-conditioned channel estimate, since every subcarrier is excited with equal magnitude.

Zadoff-Chu and CAZAC Equations

Zadoff-Chu sequence (length N, root u, odd N):
x_u[n] = exp(−j × π × u × n × (n+1) / N), n = 0 … N−1

Constant amplitude:
|x_u[n]| = 1 for all n → PAPR = 0 dB

Zero periodic autocorrelation:
R[τ] = ∑_n x[n] × x*[(n+τ) mod N] = N·δ[τ]

Bounded cross-correlation (prime N):
|R_u,v[τ]| = √N for all τ when u ≠ v (prime N)

Where N = sequence length, u = root index (coprime to N), τ = cyclic shift, x* = complex conjugate, δ = Kronecker delta. Example: N = 839 for the LTE PRACH preamble yields cross-correlation magnitude √839 ≈ 29, i.e. about 1/29 of the N = 839 autocorrelation peak.

CAZAC Sequence Families Compared

Sequence family	Alphabet	Periodic autocorr.	Cross-correlation	PAPR	Typical use
Zadoff-Chu	Polyphase (unit circle)	Ideal (impulse)	√N (prime N)	0 dB	LTE/5G PSS, PRACH, SRS
Frank-Zadoff	Polyphase, N = L²	Ideal (impulse)	Moderate	0 dB	Radar pulse compression
Chu polyphase	Polyphase	Ideal (impulse)	Bounded	0 dB	Channel sounding, test
m-sequence (PN)	Binary ±1	−1 sidelobes	Variable	0 dB (BPSK)	Spread spectrum, scrambling
Golay complementary	Binary ±1	Sum is impulse	Low (pairwise)	0 dB (BPSK)	802.11ad preamble

Common Questions

Frequently Asked Questions

How is a Zadoff-Chu sequence constructed and why is it CAZAC?

A length-N Zadoff-Chu sequence is x_u[n] = exp(−jπu·n(n+1)/N) for odd N, with root u coprime to N. Every sample lies on the unit circle, giving constant amplitude, and the quadratic phase makes the periodic autocorrelation an impulse, giving zero autocorrelation at all nonzero shifts. For prime N the cross-correlation between roots is bounded by √N, which is why LTE assigns different roots to neighboring cells.

Why do CAZAC sequences have a peak-to-average power ratio of 0 dB?

Because every time sample has identical magnitude, the instantaneous power never deviates from the mean, so PAPR is exactly 0 dB. The flat envelope lets a power amplifier operate near saturation without the back-off that high-PAPR OFDM payloads demand, improving efficiency and uplink range. This is why 3GPP chose a Zadoff-Chu CAZAC sequence for the PRACH preamble and uplink reference signals; after DFT precoding and pulse shaping the continuous waveform shows only a small, bounded PAPR increase.

How does the ideal autocorrelation improve timing and channel estimation?

An impulse-like autocorrelation gives a correlator one sharp peak at perfect alignment and no sidelobes, enabling sample-accurate frame and symbol timing, refinable to sub-sample precision by peak interpolation, and clean resolution of individual multipath taps. In the frequency domain the constant magnitude excites every subcarrier equally, so dividing the received spectrum by the known sequence yields a well-conditioned least-squares channel estimate with uniform noise across the band. The LTE primary synchronization signal relies on exactly this behavior for initial cell search.

Related Terms

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