Amplitude-shift keying is a form of
modulationIn electronics and telecommunications, modulation is the process of varying one or more properties of a high-frequency periodic waveform, called the carrier signal, with a modulating signal which typically contains information to be transmitted...
that represents
digitalA digital system is a data technology that uses discrete values. By contrast, non-digital systems use a continuous range of values to represent information...
dataThe term data refers to qualitative or quantitative attributes of a variable or set of variables. Data are typically the results of measurements and can be the basis of graphs, images, or observations of a set of variables. Data are often viewed as the lowest level of abstraction from which...
as variations in the
amplitudeAmplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
of a
carrier waveIn telecommunications, a carrier wave or carrier is a waveform that is modulated with an input signal for the purpose of conveying information. This carrier wave is usually a much higher frequency than the input signal...
.
Any digital modulation scheme uses a finite number of distinct signals to represent digital data. ASK uses a finite number of amplitudes, each assigned a unique pattern of
binary digitA bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...
s. Usually, each amplitude encodes an equal number of bits. Each pattern of bits forms the symbol that is represented by the particular amplitude. The demodulator, which is designed specifically for the symbol-set used by the modulator, determines the amplitude of the received signal and maps it back to the symbol it represents, thus recovering the original data.
FrequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
and
phasePhase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...
of the carrier are kept constant.
Like
AMAmplitude modulation is a technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. AM works by varying the strength of the transmitted signal in relation to the information being sent...
, ASK is also linear and sensitive to atmospheric noise, distortions, propagation conditions on different routes in PSTN, etc. Both ASK modulation and demodulation processes are relatively inexpensive. The ASK technique is also commonly used to transmit digital data over optical fiber. For LED transmitters, binary 1 is represented by a short pulse of light and binary 0 by the absence of light. Laser transmitters normally have a fixed "bias" current that causes the device to emit a low light level. This low level represents binary 0, while a higher-amplitude lightwave represents binary 1.
Encoding
The simplest and most common form of ASK operates as a switch, using the presence of a carrier wave to indicate a
binaryThe binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
one and its absence to indicate a binary zero. This type of modulation is called
on-off keyingOn-off keying the simplest form of amplitude-shift keying modulation that represents digital data as the presence or absence of a carrier wave. In its simplest form, the presence of a carrier for a specific duration represents a binary one, while its absence for the same duration represents a...
, and is used at
radio frequenciesRadio frequency is a rate of oscillation in the range of about 3 kHz to 300 GHz, which corresponds to the frequency of radio waves, and the alternating currents which carry radio signals...
to transmit
Morse codeMorse code is a method of transmitting textual information as a series of on-off tones, lights, or clicks that can be directly understood by a skilled listener or observer without special equipment...
(referred to as
continuous waveA continuous wave or continuous waveform is an electromagnetic wave of constant amplitude and frequency; and in mathematical analysis, of infinite duration. Continuous wave is also the name given to an early method of radio transmission, in which a carrier wave is switched on and off...
operation),
More sophisticated encoding schemes have been developed which represent data in groups using additional amplitude levels. For instance, a four-level encoding scheme can represent two
bitA bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...
s with each shift in amplitude; an eight-level scheme can represent three bits; and so on. These forms of amplitude-shift keying require a high
signal-to-noise ratioSignal-to-noise ratio is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. It is defined as the ratio of signal power to the noise power. A ratio higher than 1:1 indicates more signal than noise...
for their recovery, as by their nature much of the signal is transmitted at reduced power.
Here is a diagram showing the ideal model for a transmission system using an ASK modulation:
It can be divided into three blocks. The first one represents the transmitter, the second one is a linear model of the effects of the channel, the third one shows the structure of the receiver. The following notation is used:
- ht(f) is the carrier signal for the transmission
- hc(f) is the impulse response of the channel
- n(t) is the noise introduced by the channel
- hr(f) is the filter at the receiver
- L is the number of levels that are used for transmission
- Ts is the time between the generation of two symbols
Different symbols are represented with different voltages. If the maximum allowed value for the voltage is
A, then all the possible values are in the range [−
A,
A] and they are given by:
the difference between one voltage and the other is:
Considering the picture, the symbols
v[
n] are generated randomly by the source
S, then the
impulse generator creates impulses with an area of
v[
n]. These impulses are sent to the filter
ht to be sent through the channel. In other words, for each symbol a different carrier wave is sent with the relative amplitude.
Out of the transmitter, the signal
s(
t) can be expressed in the form:
In the receiver, after the filtering through
hr (
t) the signal is:
where we use the notation:
where * indicates the
convolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
between two signals. After the
A/D conversionAn analog-to-digital converter is a device that converts a continuous quantity to a discrete time digital representation. An ADC may also provide an isolated measurement...
the signal
z[
k] can be expressed in the form:
In this relationship, the second term represents the symbol to be extracted. The others are unwanted: the first one is the effect of noise, the second one is due to the
intersymbol interferenceIn telecommunication, intersymbol interference is a form of distortion of a signal in which one symbol interferes with subsequent symbols. This is an unwanted phenomenon as the previous symbols have similar effect as noise, thus making the communication less reliable...
.
If the filters are chosen so that
g(
t) will satisfy the
Nyquist ISI criterionIn communications, the Nyquist ISI criterion describes the conditions which, when satisfied by a communication channel, result in no intersymbol interference or ISI...
, then there will be no intersymbol interference and the value of the sum will be zero, so:
the transmission will be affected only by noise.
Probability of error
The
probability density functionIn probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
of having an error of a given size can be modelled by a
Gaussian function; the mean value will be the relative sent value, and its
varianceIn probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
will be given by:
where
is the
spectral densityIn statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...
of the noise within the band and
Hr (
f) is the
continuous Fourier transformThe Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum. For instance, the transform of a musical chord made up of pure notes is a mathematical representation of the amplitudes of the individual notes that make...
of the impulse response of the filter
hr (
f).
The
probability of making an errorIn statistics, the term "error" arises in two ways. Firstly, it arises in the context of decision making, where the probability of error may be considered as being the probability of making a wrong decision and which would have a different value for each type of error...
is given by:
where, for example,
is the
conditional probabilityIn probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...
of making an error given that a symbol
v0 has been sent and
is the probability of sending a symbol
v0.
If the probability of sending any symbol is the same, then:
If we represent all the probability density functions on the same plot against the possible value of the voltage to be transmitted, we get a picture like this (the particular case of
L = 4 is shown):
The
probability of making an errorIn statistics, the term "error" arises in two ways. Firstly, it arises in the context of decision making, where the probability of error may be considered as being the probability of making a wrong decision and which would have a different value for each type of error...
after a single symbol has been sent is the area of the Gaussian function falling under the functions for the other symbols. It is shown in cyan for just one of them. If we call
P+ the area under one side of the Gaussian, the sum of all the areas will be:
. The total probability of making an error can be expressed in the form:
We have now to calculate the value of
P+.
In order to do that, we can move the origin of the reference wherever we want: the area below the function will not change. We are in a situation like the one shown in the following picture:
it does not matter which Gaussian function we are considering, the area we want to calculate will be the same. The value we are looking for will be given by the following integral:
where erfc is the complementary error function. Putting all these results together, the probability to make an error is:
from this formula we can easily understand that the probability to make an error decreases if the maximum amplitude of the transmitted signal or the amplification of the system becomes greater; on the other hand, it increases if the number of levels or the power of noise becomes greater.
This relationship is valid when there is no intersymbol interference, i.e.
g(
t) is a
Nyquist functionIn communications, the Nyquist ISI criterion describes the conditions which, when satisfied by a communication channel, result in no intersymbol interference or ISI...
.
See also
- Frequency-shift keying
Frequency-shift keying is a frequency modulation scheme in which digital information is transmitted through discrete frequency changes of a carrier wave. The simplest FSK is binary FSK . BFSK uses a pair of discrete frequencies to transmit binary information. With this scheme, the "1" is called...
- Phase-shift keying
Phase-shift keying is a digital modulation scheme that conveys data by changing, or modulating, the phase of a reference signal ....
- Nyquist ISI criterion
In communications, the Nyquist ISI criterion describes the conditions which, when satisfied by a communication channel, result in no intersymbol interference or ISI...
- Intersymbol interference
In telecommunication, intersymbol interference is a form of distortion of a signal in which one symbol interferes with subsequent symbols. This is an unwanted phenomenon as the previous symbols have similar effect as noise, thus making the communication less reliable...
External links