VoIP Basics

Nyquist Theorem -- Sampling Rate Versus Bandwidth

The Nyquist theorem states that a signal must be sampled at least twice as fast as the bandwidth of the signal to accurately reconstruct the waveform; otherwise, the high-frequency content will alias at a frequency inside the spectrum of interest (passband). An alias is a false lower frequency component that appears in sampled data acquired at too low a sampling rate. The following figure shows a 5 MHz sine wave digitized by a 6 MS/s ADC. The dotted line indicates the aliased signal recorded by the ADC at that sample rate.

The 5 MHz frequency aliases back in the passband, falsely appearing as a 1 MHz sine wave. To prevent aliasing in the passband, a lowpass filter limits the frequency content of the input signal above the Nyquist rate.

[Sampling Theorem]:

A signal or function is bandlimited if it contains no energy at frequencies higher than some bandlimit or bandwidth. A signal that is bandlimited is constrained in how rapidly it changes in time, and therefore how much detail it can convey in an interval of time. The sampling theorem asserts that the uniformly spaced discrete samples are a complete representation of the signal if this bandwidth is less than half the sampling rate.

To formalize these concepts, let represent a continuous-time signal and be the continuous Fourier transform of that signal (which exists if is square-integrable):

The signal is bandlimited to a one-sided baseband bandwidth if:

for all $|f| > B ,$

Then the sufficient condition for exact reconstructability from samples at a uniform sampling rate (in samples per unit time) or equivalently, is called the Nyquist rate and is a property of the bandlimited signal, while is called the Nyquist frequency and is a property of this sampling system.

The time interval between successive samples is referred to as the sampling interval and the samples of are denoted by (integers).

The sampling theorem leads to a procedure for reconstructing the original from the samples and states sufficient conditions for such a reconstruction to be exact.

The sampling process

The theorem describes two processes in signal processing: a sampling process, in which a continuous time signal is converted to a discrete time signal, and a reconstruction process, in which the original continuous signal is recovered from the discrete time signal. The continuous signal varies over time (or space in a digitized image, or another independent variable in some other application) and the sampling process is performed by measuring the continuous signal's value every T units of time (or space), which is called the sampling interval. In practice, for signals that are a function of time, the sampling interval is typically quite small, on the order of milliseconds, microseconds, or less. This results in a sequence of numbers, called samples, to represent the original signal. Each sample value is associated with the instant in time when it was measured. The reciprocal of the sampling interval (1/T) is the sampling frequency denoted f_s, which is measured in samples per unit of time. If T is expressed in seconds, then f_s is expressed in Hz.

Reconstruction of the original signal is an interpolation process that mathematically defines a continuous-time signal x(t) from the discrete samples x[n] and at times in between the sample instants nT.

The normalized sinc function: sin(πx) / (πx) ... showing the central peak at x= 0, and zero-crossings at the other integer values of x.

The procedure: Each sample value is multiplied by the sinc function scaled so that the zero-crossings of the sinc function occur at the sampling instants and that the sinc function's central point is shifted to the time of that sample, nT. All of these shifted and scaled functions are then added together to recover the original signal. The scaled and time-shifted sinc functions are continuous making the sum of these also continuous, so the result of this operation is a continuous signal. This procedure is represented by the Whittaker–Shannon interpolation formula.

The condition: The signal obtained from this reconstruction process can have no frequencies higher than one-half the sampling frequency. According to the theorem, the reconstructed signal will match the original signal provided that the original signal contains no frequencies at or above this limit. This condition is called the Nyquist criterion, or sometimes the Raabe condition.

If the original signal contains a frequency component equal to one-half the sampling rate, the condition is not satisfied. The resulting reconstructed signal may have a component at that frequency, but the amplitude and phase of that component generally will not match the original component.

This reconstruction or interpolation using sinc functions is not the only interpolation scheme. Indeed, it is impossible in practice because it requires summing an infinite number of terms. However, it is the interpolation method that in theory exactly reconstructs any given bandlimited x(t) with any bandlimit B < 1/2T); any other method that does so is formally equivalent to it.

Practical considerations

A few consequences can be drawn from the theorem:

If the highest frequency B in the original signal is known, the theorem gives the lower bound on the sampling frequency for which perfect reconstruction can be assured. This lower bound to the sampling frequency, 2B, is called the Nyquist rate.

If instead the sampling frequency is known, the theorem gives us an upper bound for frequency components, B<f_s/2, of the signal to allow for perfect reconstruction. This upper bound is the Nyquist frequency, denoted f_N.

Both of these cases imply that the signal to be sampled must be bandlimited; that is, any component of this signal which has a frequency above a certain bound should be zero, or at least sufficiently close to zero to allow us to neglect its influence on the resulting reconstruction. In the first case, the condition of bandlimitation of the sampled signal can be accomplished by assuming a model of the signal which can be analysed in terms of the frequency components it contains; for example, sounds that are made by a speaking human normally contain very small frequency components at or above 10 kHz and it is then sufficient to sample such an audio signal with a sampling frequency of at least 20 kHz. For the second case, we have to assure that the sampled signal is bandlimited such that frequency components at or above half of the sampling frequency can be neglected. This is usually accomplished by means of a suitable low-pass filter; for example, if it is desired to sample speech waveforms at 8 kHz, the signals should first be lowpass filtered to below 4 kHz.

In practice, neither of the two statements of the sampling theorem described above can be completely satisfied, and neither can the reconstruction formula be precisely implemented. The reconstruction process that involves scaled and delayed sinc functions can be described as ideal. It cannot be realized in practice since it implies that each sample contributes to the reconstructed signal at almost all time points, requiring summing an infinite number of terms. Instead, some type of approximation of the sinc functions, finite in length, has to be used. The error that corresponds to the sinc-function approximation is referred to as interpolation error. Practical digital-to-analog converters produce neither scaled and delayed sinc functions nor ideal impulses (that if ideally low-pass filtered would yield the original signal), but a sequence of scaled and delayed rectangular pulses. This practical piecewise-constant output can be modeled as a zero-order hold filter driven by the sequence of scaled and delayed dirac impulses referred to in the mathematical basis section below. A shaping filter is sometimes used after the DAC with zero-order hold to make a better overall approximation.

Furthermore, in practice, a sampled signal that is "time-limited", or finite length, can never be fully bandlimited. This means that even if an ideal reconstruction could be made, the reconstructed signal would not be exactly the original signal. The error that corresponds to the failure of bandlimitation is referred to as aliasing.

The sampling theorem does not say what happens when the conditions and procedures are not exactly met, but its proof suggests an analytical framework in which the non-ideality can be studied. A designer of a system that deals with sampling and reconstruction processes needs a thorough understanding of the signal to be sampled, in particular its frequency content, the sampling frequency, how the signal is reconstructed in terms of interpolation, and the requirement for the total reconstruction error, including aliasing and interpolation error. These properties and parameters may need to be carefully tuned in order to obtain a useful system.

Aliasing

Hypothetical spectrum of a properly sampled bandlimited signal (blue) and images (green) that do not overlap. A "brick-wall" low-pass filter can remove the images and leave the original spectrum, thus recovering the original signal from the samples.

If the sampling condition is not satisfied, then frequencies will overlap; that is, frequencies above half the sampling rate will be reconstructed as, and appear as, frequencies below half the sampling rate. The resulting distortion is called aliasing; the reconstructed signal is said to be an alias of the original signal, in the sense that it has the same set of sample