Zadoff–Chu sequence
Encyclopedia
A Zadoff–Chu sequence is a complex-valued
mathematical sequence
which, when applied to radio
signals, gives rise to an electromagnetic signal of constant amplitude, whereby cyclically shifted versions of the sequence imposed on a signal result in zero cross-correlation with one another at the receiver. A generated Zadoff–Chu sequence that has not been shifted is known as a "root sequence".
The sequence then exhibits the useful property that cyclically shifted versions of itself are orthogonal
to one another, provided, that is, that each cyclic shift, when viewed within the time domain
of the signal, is greater than the combined propagation delay and multi-path delay-spread of that signal between the transmitter and receiver.
The complex value at each position (n) of each root Zadoff–Chu sequence (u) given by
where
Zadoff–Chu sequence is known as a CAZAC sequence (constant amplitude zero autocorrelation waveform
).
It is named after Solomon A. Zadoff and D. C. Chu.
with period
if 
is prime.
2. Given
is prime, Discrete Fourier Transform
of Zadoff–Chu sequence is another Zadoff–Chu sequence conjugated, scaled and time scaled.
3. The autocorrelation
of a prime length Zadoff–Chu sequence with a cyclically shifted version of itself also has zero auto-correlation, i.e., it is non-zero only at one instant which corresponds to the cyclic shift.
4. The cross correlation between two prime length Zadoff–Chu sequences, i.e. different
, is constant 
eNodeB and multiplying their transmissions by their respective codes, the cross-correlation
of simultaneous eNodeB transmissions is reduced, thus reducing intra-cell interference and uniquely identifying eNodeB transmissions. Zadoff-Chu sequence improve over the Walsh–Hadamard codes used in UMTS because they result in a constant-amplitude output signal, reducing the cost and complexity of the radio's power amplifier.
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
mathematical sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
which, when applied to radio
Radio
Radio is the transmission of signals through free space by modulation of electromagnetic waves with frequencies below those of visible light. Electromagnetic radiation travels by means of oscillating electromagnetic fields that pass through the air and the vacuum of space...
signals, gives rise to an electromagnetic signal of constant amplitude, whereby cyclically shifted versions of the sequence imposed on a signal result in zero cross-correlation with one another at the receiver. A generated Zadoff–Chu sequence that has not been shifted is known as a "root sequence".
The sequence then exhibits the useful property that cyclically shifted versions of itself are orthogonal
Orthogonality
Orthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...
to one another, provided, that is, that each cyclic shift, when viewed within the time domain
Time domain
Time domain is a term used to describe the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the case of continuous time, or at various...
of the signal, is greater than the combined propagation delay and multi-path delay-spread of that signal between the transmitter and receiver.
The complex value at each position (n) of each root Zadoff–Chu sequence (u) given by
where
Zadoff–Chu sequence is known as a CAZAC sequence (constant amplitude zero autocorrelation waveform
Constant amplitude zero autocorrelation waveform
A Constant Amplitude Zero AutoCorrelation waveform is a periodic complex-valued signal with modulus one and out-of-phase periodic autocorrelation equal to zero. CAZAC sequences find application in wireless communication systems, for example in LTE for synchronisation of mobile phones with base...
).
It is named after Solomon A. Zadoff and D. C. Chu.
Properties of Zadoff-Chu sequences
1. They are periodicPeriodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
with period
if is prime.
2. Given
is prime, Discrete Fourier TransformDiscrete Fourier transform
In mathematics, the discrete Fourier transform is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function...
of Zadoff–Chu sequence is another Zadoff–Chu sequence conjugated, scaled and time scaled.
-
where 
is the multiplicative inverse of u modulo 
.
3. The autocorrelation
Autocorrelation
Autocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them...
of a prime length Zadoff–Chu sequence with a cyclically shifted version of itself also has zero auto-correlation, i.e., it is non-zero only at one instant which corresponds to the cyclic shift.
4. The cross correlation between two prime length Zadoff–Chu sequences, i.e. different
, is constant Usages
Zadoff–Chu sequences are used in the 3GPP LTE Long Term Evolution air interface in the Primary Synchronization Signal (PSS) (so-called primary synchronization channel), random access preamble (PRACH), HARQ ACK/NACK responses (PUCCH) and sounding reference signals (SRS). By assigning orthogonal Zadoff-Chu sequences to each LTE3GPP Long Term Evolution
3GPP Long Term Evolution, usually referred to as LTE, is a standard for wireless communication of high-speed data for mobile phones and data terminals. It is based on the GSM/EDGE and UMTS/HSPA network technologies, increasing the capacity and speed using new modulation techniques...
eNodeB and multiplying their transmissions by their respective codes, the cross-correlation
Cross-correlation
In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long-duration signal for a shorter, known feature...
of simultaneous eNodeB transmissions is reduced, thus reducing intra-cell interference and uniquely identifying eNodeB transmissions. Zadoff-Chu sequence improve over the Walsh–Hadamard codes used in UMTS because they result in a constant-amplitude output signal, reducing the cost and complexity of the radio's power amplifier.