Flat Top Sampling
Pulses of the type shown in Fig1 (a) with tops contoured to follow the waveform of the signal, are actually not frequently employed.
Instead flat-topped pulses are customarily used, as shown in Fig. 2(a).
A flat-topped pulse has a constant amplitude established by the sample value of the signal at some point within the pulse interval.
In Fig 2 (a) we have arbitrarily sampled the signal at the beginning of the pulse.
In sampling of this type the baseband signal m(t) cannot be recovered exactly by simply passing the samples through an ideal low-pass filter.
However, the distortion need not be large.
Flat top sampling has the merit that it simplifies the design of the electronic circuitry used to perform the sampling operation.
To show the extent of the distortion, consider the signal m(t) having a Fourier transform M(jω).
We have seen (sampling Theorem and Low Pass Signal, Sampling of Bandpass Signal, PULSE AMPLITUDE MODULATION And Channel Bandwidth for PAM )
how to deduce the transform of the sampled signal, when the sampling is instantaneous.
The transform of the sampled signal for flat-top sampling is determined by that the flat-top pulse can be generated
by passing the instantaneously sampled signal through a network which broadens a pulse of duration dt (an impulse) into a pulse of duration τ.
The transform of a pulse of unit amplitude and width dt is
F [impulse of strength dt at t=0]=dt….(1)
The transform of a pulse of unit amplitude and width τ is
F [ pulse, amplitude =1, extending from t
= -τ/2 to t=τ/2] = τ sin ( ωτ/ 2) / ωτ/ 2…(2)
Hence, the transfer function of the network shown in Fig. 2(b) is required to be
H(jω) = τ /dt . sin(ωτ/2)/ωτ/ 2 ….(3)
Let the signal m(t), with transform M(jω), be bandlimited to fm and be sampled at the Nyquist rate or faster.
Then in the range 0 to fm the transform of the flat-topped sampled signal is given by the product H(jω)M(jω) or, from Eqs (1), (2), and (3)
F [flat-topped sampled m(t)]
= τ/ Ts . sin(ωτ/2)/ωτ/ 2 M (jω) …..(4)
0≤ f≤ fm
To illustrate the effect of flat top sampling, we consider for simplicity that the signal m(t) has a flat spectral density equal to Mo over its entire rang from 0 to fm ,
as is shown in Fig. 3a.
The form of the transform of the instantaneously sampled signal is shown in Fig.3b.
The sampling frequency fs = 1/Ts, is assumed large enough to allow for a guard band between the spectrum of the baseband signal and the DSB-SC signal with carrier fs.
The spectrum of the flat-topped sampled signal is shown in Fig. 3d.
We are, of course, interested only in the part of the spectrum in the range 0 to fm .
If in this range, the spectra of the sampled signal and the original signal are identical,
than the original signal may be recovered by a low-pass filter as has already been discussed.
We observe, however that such is not the case and that, as a result, distortion will result.
This distortion results from the fact that the original signal was “observed through a finite rather than an infinitesimal time “aperture” and is hence referred to as aperture effect distortion.
The distortion results from the fact that the spectrum is multiplied by the sampling function Sa(x)≡ (sin x)/x (with x=ωτ/2).
The magnitude of the sampling function falls off slowly with increasing x in the neighborhood of x = 0 and does not fall off sharply until we approach x= π,
at which point Sa(x)= 0.
To minimize the distortion due to the aperture effect, it is advantageous to arrange that x = π correspond to a frequency very large in comparison with fm.
Since x= πfτ, the frequency fo corresponding to x= π is fo =1/τ.
If fo >>fm or, correspondingly, if τ<<1,/fm, the aperture distortion will be small.
The distortion becomes progressively smaller with decreasing τ.
And, of course as τ→0 (instantaneous sampling), the distortion similarly approaches zero.
Equalization¹, ²
As in the case of natural sampling, so also in the present case of flat-top sampling, it is advantageous to make ras large as practicable for the sake of increasing the amplitude of the output signal.
If, in a particular case, it should happen that the consequent distortion is not acceptable,
it may be corrected by including an equalizer in cascade with the output low-pass filter.
An equalizer, in the present instance, is a passive network whose transfer function has a frequency dependence of the form x/sinx, that is, a form inverse to the form of H(jω) given in Eq. (3).
The equalizer in combination with the aperture effect will then yield a flat overall transfer characteristic between the original baseband signal and the output at the receiving end of the system.
The equalizer x/sinx cannot be exactly synthesized, but can be approximated.
If N signals are multiplexed, τ≤ 1/2FmN, and hence for large N τ<< 1/fm, and x/sin x ≅1.
In this case the equalizer is not needed as negligible distortion results.
Multiplying of two 8 bit number–
Addressing modes of 8085 microprocessor
Sampling Theorem and Low Pass Signal
