Phase-shift keying


Phase-shift keying is a digital modulation process which conveys data by changing the phase of a constant frequency carrier wave. The modulation is accomplished by varying the sine and cosine inputs at a precise time. It is widely used for wireless LANs, RFID and Bluetooth communication.
Any digital modulation scheme uses a finite number of distinct signals to represent digital data. PSK uses a finite number of phases, each assigned a unique pattern of binary digits. Usually, each phase encodes an equal number of bits. Each pattern of bits forms the symbol that is represented by the particular phase. The demodulator, which is designed specifically for the symbol-set used by the modulator, determines the phase of the received signal and maps it back to the symbol it represents, thus recovering the original data. This requires the receiver to be able to compare the phase of the received signal to a reference signal such a system is termed coherent.
CPSK requires a complicated demodulator, because it must extract the reference wave from the received signal and keep track of it, to compare each sample to. Alternatively, the phase shift of each symbol sent can be measured with respect to the phase of the previous symbol sent. Because the symbols are encoded in the difference in phase between successive samples, this is called differential phase-shift keying . DPSK can be significantly simpler to implement than ordinary PSK, as it is a 'non-coherent' scheme, i.e. there is no need for the demodulator to keep track of a reference wave. A trade-off is that it has more demodulation errors.

Introduction

There are three major classes of digital modulation techniques used for transmission of digitally represented data:
All convey data by changing [|some] aspect of a base signal, the carrier wave, in response to a data signal. In the case of PSK, the phase is changed to represent the data signal. There are two fundamental ways of utilizing the phase of a signal in this way:
  • By viewing the phase itself as conveying the information, in which case the demodulator must have a reference signal to compare the received signal's phase against; or
  • By viewing the change in the phase as conveying information differential schemes, some of which do not need a reference carrier.
A convenient method to represent M-ary transmission PSK schemes is on a constellation diagram. This shows the points in the complex plane where, in this context, the real and imaginary axes are termed the in-phase and quadrature axes respectively due to their 90° separation. Such a representation on perpendicular axes lends itself to straightforward implementation. The amplitude of each point along the in-phase axis is used to modulate a cosine wave and the amplitude along the quadrature axis to modulate a sine wave. By convention, in-phase modulates cosine and quadrature modulates sine.
In PSK, the constellation points chosen are usually positioned with uniform angular spacing around a circle. This gives maximum phase-separation between adjacent points and thus the best immunity to corruption. They are positioned on a circle so that they can all be transmitted with the same energy. In this way, the moduli of the complex numbers they represent will be the same and thus so will the amplitudes needed for the cosine and sine waves. Two common examples are "binary phase-shift keying" which uses two phases, and "quadrature phase-shift keying" which uses four phases, although any number of phases may be used. Since the data to be conveyed are usually binary, the PSK scheme is usually designed with the number of constellation points being a power of two.

Definitions

For determining error-rates mathematically, some definitions will be needed:
will give the probability that a single sample taken from a random process with zero-mean and unit-variance Gaussian probability density function will be greater or equal to. It is a scaled form of the complementary Gaussian error function:
The error rates quoted here are those in additive white Gaussian noise. These error rates are lower than those computed in fading channels, hence, are a good theoretical benchmark to compare with.

Binary phase-shift keying (BPSK)

BPSK is the simplest form of phase shift keying. It uses two phases which are separated by 180° and so can also be termed 2-PSK. It does not particularly matter exactly where the constellation points are positioned, and in this figure they are shown on the real axis, at 0° and 180°. Therefore, it handles the highest noise level or distortion before the demodulator reaches an incorrect decision. That makes it the most robust of all the PSKs. It is, however, only able to modulate at 1bit/symbol and so is unsuitable for high data-rate applications.
In the presence of an arbitrary phase-shift introduced by the communications channel, the demodulator is unable to tell which constellation point is which. As a result, the data is often [|differentially encoded] prior to modulation.
BPSK is functionally equivalent to 2-QAM modulation.

Implementation

The general form for BPSK follows the equation:
This yields two phases, 0 and π.
In the specific form, binary data is often conveyed with the following signals:
where f is the frequency of the base band.
Hence, the signal space can be represented by the single basis function
where 1 is represented by and 0 is represented by. This assignment is arbitrary.
This use of this basis function is shown at the [|end of the next section] in a signal timing diagram. The topmost signal is a BPSK-modulated cosine wave that the BPSK modulator would produce. The bit-stream that causes this output is shown above the signal. After modulation, the base band signal will be moved to the high frequency band by multiplying.

Bit error rate

The bit error rate of BPSK under additive white Gaussian noise can be calculated as:
Since there is only one bit per symbol, this is also the symbol error rate.

Quadrature phase-shift keying (QPSK)

Sometimes this is known as quadriphase PSK, 4-PSK, or 4-QAM. QPSK uses four points on the constellation diagram, equispaced around a circle. With four phases, QPSK can encode two bits per symbol, shown in the diagram with Gray coding to minimize the bit error rate sometimes misperceived as twice the BER of BPSK.
The mathematical analysis shows that QPSK can be used either to double the data rate compared with a BPSK system while maintaining the same bandwidth of the signal, or to maintain the data-rate of BPSK but halving the bandwidth needed. In this latter case, the BER of QPSK is exactly the same as the BER of BPSK and believing differently is a common confusion when considering or describing QPSK. The transmitted carrier can undergo numbers of phase changes.
Given that radio communication channels are allocated by agencies such as the Federal Communications Commission giving a prescribed bandwidth, the advantage of QPSK over BPSK becomes evident: QPSK transmits twice the data rate in a given bandwidth compared to BPSK - at the same BER. The engineering penalty that is paid is that QPSK transmitters and receivers are more complicated than the ones for BPSK. However, with modern electronics technology, the penalty in cost is very moderate.
As with BPSK, there are phase ambiguity problems at the receiving end, and differentially encoded QPSK is often used in practice.

Implementation

The implementation of QPSK is more general than that of BPSK and also indicates the implementation of higher-order PSK. Writing the symbols in the constellation diagram in terms of the sine and cosine waves used to transmit them:
This yields the four phases π/4, 3π/4, 5π/4 and 7π/4 as needed.
This results in a two-dimensional signal space with unit basis functions
The first basis function is used as the in-phase component of the signal and the second as the quadrature component of the signal.
Hence, the signal constellation consists of the signal-space 4 points
The factors of 1/2 indicate that the total power is split equally between the two carriers.
Comparing these basis functions with that for BPSK shows clearly how QPSK can be viewed as two independent BPSK signals. Note that the signal-space points for BPSK do not need to split the symbol energy over the two carriers in the scheme shown in the BPSK constellation diagram.
QPSK systems can be implemented in a number of ways. An illustration of the major components of the transmitter and receiver structure are shown below.

Probability of error

Although QPSK can be viewed as a quaternary modulation, it is easier to see it as two independently modulated quadrature carriers. With this interpretation, the even bits are used to modulate the in-phase component of the carrier, while the odd bits are used to modulate the quadrature-phase component of the carrier. BPSK is used on both carriers and they can be independently demodulated.
As a result, the probability of bit-error for QPSK is the same as for BPSK:
However, in order to achieve the same bit-error probability as BPSK, QPSK uses twice the power.
The symbol error rate is given by:
If the signal-to-noise ratio is high the probability of symbol error may be approximated:
The modulated signal is shown below for a short segment of a random binary data-stream. The two carrier waves are a cosine wave and a sine wave, as indicated by the signal-space analysis above. Here, the odd-numbered bits have been assigned to the in-phase component and the even-numbered bits to the quadrature component. The total signal the sum of the two components is shown at the bottom. Jumps in phase can be seen as the PSK changes the phase on each component at the start of each bit-period. The topmost waveform alone matches the description given for BPSK above.


The binary data that is conveyed by this waveform is: 11000110.
  • The odd bits, highlighted here, contribute to the in-phase component: 1010
  • The even bits, highlighted here, contribute to the quadrature-phase component: 1001