Realistic Sampling of Signals

Table of Contents
1. Practical implementation using complementary pulse trains
2. Fourier series representation of a periodic pulse train
3. Fourier transform of the sampled signal
4. Generalisation: sampling by any periodic waveform
5. Applications and practical observations
6. Summary
View more

Realistic sampling of signals:

Sampling a continuous-time signal in practice is realised by multiplying the signal with a train of pulses. Ideal impulses (Dirac deltas) are a mathematical abstraction and cannot be produced exactly in hardware. In real systems we use a periodic pulse train (for example rectangular pulses or saw-tooth derived gating pulses) generated by time-base circuits such as those used in a cathode-ray oscilloscope (CRO). Multiplication by such a pulse train produces a sequence of gated copies of the continuous-time signal; this is referred to here as realistic sampling.

Practical implementation using complementary pulse trains

A common hardware method uses two complementary pulse trains synchronized so that when one is high the other is low. By gating the input signal with one of these pulse trains through an electronic multiplier or switch, we obtain a sampled output that is non-zero only during the ON intervals of the chosen pulse train.

Consider the schematic where the signal x(t) is multiplied by a pulse train p1(t) while the complementary train p2(t) is low during those intervals. The multiplication output is present only when p1(t) is ON; when p2(t) is ON the output is zero. This is an implementable version of the ideal mathematical sampling operation.

In this arrangement the sampled signal is the product x(t) p(t), where p(t) is a periodic pulse train with period T (sampling period) and fundamental frequency ωs = 2π/T. The pulse width (ON duration) is denoted τ, and the duty cycle is α = τ/T.

Fourier series representation of a periodic pulse train

A periodic pulse train that satisfies Dirichlet's conditions can be expanded in a complex Fourier series. If p(t) has period T and complex Fourier coefficients Ck, then

p(t) = Σ_k=-∞^∞ C_k e^{j k ωs t}, where ωs = 2π/T.

The Fourier coefficients are given by the period integral

C_k = (1/T) ∫₀^T p(t) e^{-j k ωs t} dt.

For the constant (DC) coefficient (k = 0) we get the average value of p(t) over one period:

C₀ = (1/T) ∫₀^T p(t) dt = α (for a rectangular pulse train whose amplitude during the ON interval is 1).

For a rectangular pulse of width τ centred within the period (integration limits chosen symmetrically), the kth coefficient simplifies to:

C_k = (τ/T) e^{-j k ωs τ/2} sinc( k ωs τ/2 ), where sinc(x) = sin(x)/x and ωs = 2π/T.

The magnitude of the coefficients has an envelope given by the sinc function:

The envelope |C_k| therefore decreases with |k| according to the sinc shape. The central coefficient (k = 0) equals the duty cycle α = τ/T and is the largest term in magnitude.

Effect of pulse width on Fourier coefficients

From the coefficient expression we can observe the following:

• If
  
  (i.e., τ) is small then relatively few samples contribute significant energy to the main lobe of the coefficient envelope.
• As
  
  (i.e., τ) increases, the main lobe of the sinc envelope broadens and sidelobes change accordingly.
• As
  
  (i.e., τ/T) increases toward 1, coefficients approach a constant behaviour (they tend to
  
  ) as the central lobe width tends to infinity.

In the limiting case as

Fourier transform of the sampled signal

When the continuous-time signal x(t) is multiplied by the periodic pulse train p(t), we can use the Fourier series of p(t) to find the spectrum of the sampled signal.

Write the pulse train as p(t) = Σ_k C_k e^{j k ωs t}. The sampled signal is xs(t) = x(t) p(t) = x(t) Σ_k C_k e^{j k ωs t}.

Taking the Fourier transform and using the frequency-translation property gives

X_s(ω) = Σ_k=-∞^∞ C_k X(ω - k ωs), where X(ω) is the Fourier transform of x(t).

Thus the spectrum of the sampled signal is a sum of frequency-shifted copies of the original spectrum X(ω), each copy weighted (modulated) by the corresponding Fourier series coefficient Ck.

Condition for alias-free reconstruction

If x(t) is band-limited with maximum frequency f_m (or bandwidth f_n), then reconstruction without overlap (aliasing) requires that the shifted copies do not overlap in frequency. In frequency (Hz) terms the sampling frequency fs = 1/T must satisfy

fs > 2 f_n,

so that the copies centred at integer multiples of fs are separated. Under this condition the original spectrum can be recovered theoretically by passing xs(t) through an ideal low-pass filter whose passband selects the central copy (k = 0).

The faithful reconstruction condition is therefore

where f_n denotes the highest frequency component (bandwidth) of the band-limited input signal.

Generalisation: sampling by any periodic waveform

The analysis above used a rectangular pulse train as an example, but the same reasoning applies to any periodic waveform p(t) that has a convergent Fourier series. The sampled spectrum will still be a sum of shifted copies of X(ω), with weights equal to the Fourier series coefficients of p(t). The central copy is weighted by the DC coefficient C0 (the average of p(t)).

If the periodic waveform has a non-zero average (i.e., C0 ≠ 0) and its fundamental frequency is greater than twice the bandwidth of the signal, then the central copy exists and an ideal low-pass filter can recover the original signal from the sampled representation.

If the periodic waveform has zero average (for example, certain symmetric waveforms with positive and negative lobes that cancel), then C0 = 0. In that case the central copy is suppressed and an ideal low-pass filter cannot directly recover the original signal; some other reconstruction method or preconditioning would be required.

Triangular or other pulse shapes

Replacing rectangular pulses by triangular pulses or other shapes does not change the fundamental structure: the sampled spectrum remains a sum of shifted versions of X(ω) weighted by the Fourier coefficients of the chosen pulse shape. The coefficient magnitudes (the envelope) change according to the pulse shape: triangular pulses lead to a faster roll-off of coefficients (roughly the square of the sinc envelope for rectangular pulses), which affects the relative amplitudes of the shifted spectral copies and the level of out-of-band energy.

Applications and practical observations

• Realistic sampling using pulse trains is the method implemented in practical systems such as sample-and-hold circuits, gated amplifiers and CRO time-base gates.
• The duty cycle α = τ/T controls the average energy of the central spectral copy (C0) and the distribution of energy among harmonics through the sinc envelope.
• For faithful analog-to-analog reconstruction using a simple ideal low-pass filter, ensure fs > 2 f_n and that the sampling pulse train has a non-zero average.
• Making pulses narrower (reducing τ) tends toward the impulse-train model; in the limit an impulse train has constant Fourier coefficients and produces equally weighted spectral replicas.
• Pulse shape matters for practical filtering and for the levels of higher-frequency replicas; choosing a pulse shape with favourable spectral decay reduces unwanted high-frequency content.

Summary

Sampling with a realistic periodic pulse train is modelled by multiplication in time, which yields a spectrum formed by shifted copies of the original signal spectrum weighted by the Fourier series coefficients of the pulse train. The duty cycle and shape of the pulses determine the coefficients (sinc-like envelopes for rectangular pulses). The usual aliasing condition applies: sampling frequency must exceed twice the signal bandwidth to avoid spectral overlap. Any periodic waveform with a non-zero average and sufficient fundamental frequency may be used for sampling and subsequent ideal low-pass reconstruction; if the average is zero the central spectral copy is suppressed and simple low-pass reconstruction is not possible.

Key terms: sampling, pulse train, duty cycle, Fourier series, Fourier coefficients, sinc envelope, band-limited signal, aliasing, sampling frequency, ideal low-pass reconstruction.