NationMaster - Encyclopedia: 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. Channel, in communications (sometimes called communications channel), refers to the medium used to convey information from a sender (or transmitter) to a receiver. ... In telecommunication, the term intersymbol interference has the following meanings: 1. ...

When consecutive symbols are transmitted over a channel by a linear modulation (such as ASK, QAM, etc.), the impulse response (or equivalently the frequency response) of the channel causes a transmitted symbol to be spread in the time domain. This causes intersymbol interference because the previously transmitted symbols affect the currently received symbol, thus reducing tolerance for noise. The Nyquist theorem relates this time-domain condition to an equivalent frequency-domain condition. Amplitude-shift keying (ASK) is a form of modulation which represents digital data as variations in the amplitude of a carrier wave. ... Quadrature amplitude modulation (QAM) is a modulation scheme which conveys data by changing (modulating) the amplitude of two carrier waves. ... In the language of mathematics, the impulse response of a linear transformation is the image of Diracs delta function under the transformation. ... Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ... For the Irish mythological figure, see Naoise. ... Time-domain is a term used to describe the analysis of mathematical functions, or real-life signals, with respect to time. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ...

The Nyquist criterion is closely related to the Nyquist-Shannon sampling theorem, with only a differing point of view. The Nyquist-Shannon sampling theorem is the fundamental theorem in the field of information theory, in particular telecommunications. ...

1 Nyquist criterion
2 Derivation
3 See also
4 References

Nyquist criterion

If we denote the channel impulse response as $h (t)$ , then the condition for an ISI-free response can be expressed as:

$h(n T_s) = begin{cases} 1; & n = 0 0; & n neq 0 end{cases}$

for all integer $n$ , where $T s$ is the symbol period. The Nyquist theorem says that this is equivalent to:

$sum_{k = -infty}^{+infty} H left( f - frac{k}{T_s} right) = T_s$ ,

(where $H (f)$ is the Fourier transform of $h (t)$ ). This is the Nyquist ISI criterion. In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

This criterion can be intuitively understood in the following way: frequency-shifted replicas of H(f) must add up to a constant value.

In practice this criterion is applied to baseband filtering by regarding the symbol sequence as weighted impulses (Dirac delta function). When the baseband filters in the communication system satisfy the Nyquist criterion, symbols can be transmitted over a channel with flat response within a limited frequency band, without ISI. Examples of such baseband filters are the raised-cosine filter, or the sinc filter as the ideal case. The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0, the value zero elsewhere. ... The raised-cosine filter is a particular electronic filter, frequently appearing in telecommunications systems due to its ability to minimise inter-symbol interference (ISI). ... In signal processing, a sinc filter is an idealized filter that removes all frequency components above a given bandwidth and leaves the low frequencies alone. ...

Derivation

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To derive the criterion, we first express the received signal in terms of the transmitted symbol and the channel response. Let the function h(t) be the channel impulse response, x[n] the symbols to be sent, with a symbol period of T_s; the received signal y(t) will be in the form (where noise has been ignored for simplicity): Image File history File links Broom_icon. ... In the language of mathematics, the impulse response of a linear transformation is the image of Diracs delta function under the transformation. ...

$y(t) = sum_{n = -infty}^{infty} x[n] cdot h(t - n T_s)$ .

Sampling this signal at intervals of T_s, we can express y(t) as a discrete-time equation:

$y[k] = y(k T_s) = sum_{n = -infty}^{infty} v[n] cdot h[k - n]$ .

If we write the h[0] term of the sum separately, we can express this as:

$y[k] = v[k] cdot h [0] + sum_{n neq k} v[n] cdot h[k - n]$ ,

and from this we can conclude that if a response h[n] satisfies

$h[n] = begin{cases} 1; & n = 0 0; & n neq 0 end{cases}$ ,

only one transmitted symbol has an effect on the received y[k] at sampling instants, thus removing any ISI. This is the time-domain condition for an ISI-free channel. Now we find a frequency-domain equivalent for it. We start by expressing this condition in continuous time: Time-domain is a term used to describe the analysis of mathematical functions, or real-life signals, with respect to time. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ...

$h(n T_s) = begin{cases} 1; & n = 0 0; & n neq 0 end{cases}$

for all integer $n$ . We multiply such a h(t) by a sum of Dirac delta function (impulses) $δ(t)$ separated by intervals T_s This is equivalent of sampling the response as above but using a continuous time expression. The right side of the condition can then be expressed as one impulse in the origin: The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0, the value zero elsewhere. ...

$h(t) cdot sum_{k = -infty}^{+infty} delta (t - k T_s) = delta (t)$

Fourier transforming both members of this relationship we obtain:

$H left( f right) * frac{1}{T_s}sum_{k = -infty}^{+infty} delta left( f - frac{k}{T_s} right) = 1$

and

$sum_{k = -infty}^{+infty} H left( f - frac{k}{T_s} right) = T_s$ .

This is the Nyquist ISI criterion and, if a carrier function satisfies it, then there is no ISI between the different samples.

References

John G. Proakis, "Digital Communications, 3rd Edition", McGraw-Hill Book Co., 1995. ISBN 0-07-113814-5
Behzad Razavi, "RF Microelctronics", Prentice-Hall, Inc., 1998. ISBN 0-13-887571-5

Categories: Cleanup from January 2007 | Data transmission | Digital signal processing | Control theory

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Encyclopedia > Nyquist ISI criterion

Contents

Nyquist criterion

Derivation

See also

References