communications system A communications system or communication system is a collection of individual telecommunications networks, transmission systems, relay stations, tributary stations, and terminal equipment usually capable of interconnection and interoperati ...

s, a

communication channel A communication channel refers either to a physical transmission medium such as a wire, or to a logical connection over a multiplexed medium such as a radio channel in telecommunications and computer networking. A channel is used for informa ...

may change with time. Coherence time is the time duration over which the channel impulse response is considered to be not varying. Such channel variation is much more significant in

wireless Wireless communication (or just wireless, when the context allows) is the transfer of information between two or more points without the use of an electrical conductor, optical fiber or other continuous guided medium for the transfer. The most ...

communications systems, due to

Doppler effect The Doppler effect or Doppler shift (or simply Doppler, when in context) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who d ...

Simple model

In a simple model, a signal

x(t)

transmitted at time

t_1

will be received as :

y_(t) = x(t-t_1)*h_(t),

where

h_(t)

is the channel impulse response (CIR) at time

t_1

. A signal transmitted at time

t_2

will be received as :

y_(t) = x(t-t_2)*h_(t).

Now, if

h_(t) - h_(t)

is relatively small, the channel may be considered constant within the interval

t_1

t_2

. Coherence time (

T_c

) will therefore be given by :

T_c = t_2 - t_1.

Relation with Doppler frequency

Coherence time

T_c

is the time-domain dual of Doppler spread and is used to characterize the time-varying nature of the frequency dispersiveness of the channel in the time domain. The Maximum Doppler spread and coherence time are inversely proportional to one another. That is, :

T_c\approx\frac

where

(f_m)

is the maximum Doppler spread or, maximum Doppler frequency or, maximum

Doppler shift The Doppler effect or Doppler shift (or simply Doppler, when in context) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who d ...

given by

f_m=\fracf_c

with

f_c

being the center frequency of the emitter. Coherence time is actually a statistical measure of the time duration over which the channel

impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an Dirac delta function, impulse (). More generally, an impulse ...

is essentially invariant, and quantifies the similarity of the channel response at different times. In other words, coherence time is the time duration over which two received signals have a strong potential for amplitude correlation. If the reciprocal bandwidth of the

baseband signal In telecommunications and signal processing, baseband is the range of frequencies occupied by a signal that has not been modulated to higher frequencies. Baseband signals typically originate from transducers, converting some other variable into a ...

is greater than the coherence time of the channel, then the channel will change during the transmission of the baseband message, thus causing distortion at the receiver. If the coherence time is defined as the time over which the time correlation function is above 0.5, then the coherence time is approximately, :

T_c\approx\frac

In practice, the first approximation of coherence time suggests a time duration during which a

Rayleigh fading Rayleigh fading is a statistical model for the effect of a propagation environment on a radio signal, such as that used by wireless devices. Rayleigh fading models assume that the magnitude of a signal that has passed through such a transmission me ...

signal may fluctuate wildly, and the second approximation is often too restrictive. A popular rule of thumb for modern

digital communications Data transmission and data reception or, more broadly, data communication or digital communications is the transfer and reception of data in the form of a digital bitstream or a digitized analog signal transmitted over a point-to-point or ...

is to define the coherence time as the

geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...

of the two approximate values, also known as Clarke's model; from the maximum Doppler frequency

f_m

we can obtain 50% coherence time P. Mohana Shankar, ''Introduction to Wireless Systems'', John Wiley & Sons, 2002T.S. Rappaport, ''Wireless Communications'', Prentice Hall, 2002 :

T_c=\sqrt

Usually, we use the following relation :

T_c=\sqrt\frac\simeq\frac{f_m}

References

Wireless communication systems