Sampling Theorem and Low Pass Signal
Sampling Theorem and Low Pass Signal
We consider at the outset the fundamental principle of digital communications; the sampling theorem:
Let m(t) be a signal which is bandlimited such that its highest frequency spectral component is fM .
Let the values of m(t) be determined at regular intervals separated by times Ts <= 1/2 fM , that is, the signal is periodically sampled every Ts seconds.
Then these samples m(nTs), where n is an integer, uniquely determine the signal, and the signal may be reconstructed from these samples with no distortion.
The time Ts , is called the sampling time.
Note that the theorem requires that the sampling rate be rapid enough so that at least two samples are taken during the course of the period corresponding to the highest-frequency spectral component.
We shall now prove the theorem by showing how the signal may be reconstructed from its samples.
The baseband signal m(t) which is to be sampled is shown in Fig.1a.
A periodic train of pulses S(t) of unit amplitude and of period Ts, is shown in Fig. 1b.
The pulses are arbitrarily narrow, having a width dt.
The two signals m(t) and S(t) are applied to a multiplier as shown in Fig. 1e, which then yields as an output the product S(t)m(t).
This product is seen in Fig.1d to be the signal m(t) sampled at the occurrence of each pulse.
That is, when a pulse occurs, the multiplier output has the same value as does m(t), and at all other times the multiplier output is zero.
The signal S(t) is periodic, with period Ts , and has the Fourier expansion with I = dt and T0 = Ts ]
S(t)= dt /Ts + 2dt /Ts (cos 2π t/Ts + cos 2 × 2π t/Ts + …..). ……(1)
For the case Ts = 1/2 fm the product S(t)m(t) is
S(t)m(t) = dt/Ts m(t) + dt/ Ts
[2m(t) cos 2π(2fm)t + 2m(t) cos 2π(4ƒm)t + ….]. ……. (2)
We now observe that the first term in the series is, aside from a constant factor, the signal m(t) itself.
Again, aside from a multiplying factor, the second term is the product of m(t) and a sinusoid of frequency 2fm.
This product then, as discussed gives rise to a double-sideband suppressed-carrier signal with carrier frequency 2fm.
Similarly, succeeding terms yield DSB-SC signals with carrier frequencies 4fm , 6fm etc.
Let the signal m(t) have a spectral density M(jω)= [m(t)] which is as shown in Fig. 2a.
Then m(t) is bandlimited to the frequency range below fm.
The spectrum of the first term in Eq. (2) extends from 0 to fm.
The spectrum of the second term is symmetrical about the frequency 2fm and extends from 2fm-fm =-fm to 2fm+ fm = 3fm.
Altogether the spectrum of the sampled signal has the appearance shown in Fig.2b.
Suppose then that the sampled signal is passed through an ideal low-pass filter with cutoff frequency at fm.
If the filter transmission were constant in the passband and if the cutoff were infinitely sharp at fm, the filter would pass the signal m(t) and nothing else.
The spectral pattern corresponding to Fig. 2b is shown in Fig. 3a for the case in which the sampling rate fs = 1/Ts, is larger than 2fm.
In this case there is a gap between the upper limit fm of the spectrum of the baseband signal and the lower limit of the DSB-SC spectrum centered around the carrier frequency fs> 2fm.
For this reason the low-pass filter used to select the signal m(t) need not have an infinitely sharp cutoff.
Instead, the filter attenuation may begin at fm but need not attain a high value until the frequency fs -fm.
This range from fm to fs-fm is called a guard band and is always required in practice, since a filter with infinitely sharp cutoff is, of course, not realizable.
Typically, when sampling is used in connection with voice messages on telephone lines, the voice signal is limited to fm = 3.3 kHz, while fs, is selected at 8.0 kHz.
The guard band is then 8.0-2x 3.3=1.4 kHz.
The situation depicted in Fig. 3b corresponds to the case where fs <2fm.
Here we find an overlap between the spectrum of m(t) itself and the spectrum of the DSB-SC signal centered around fs.
Accordingly, no filtering operation will allow an exact recovery of m(t). This phenomenon is known as aliasing in frequency domain.
When necessary, to avoid aliasing we use an antialiasing filter, a low pass filter that limits the frequency band of the message signal m(t) within frequency fs/2.
We have just proved the sampling theorem since we have shown that, in principle, the sampled signal can be recovered exactly when Ts <= 1/2fm .
It has also been shown why the minimum allowable sampling rate is 2fm.
This minimum sampling rate is known as the Nyquist rate.
An increase in sampling rate above the Nyquist rate increases the width of the guard band, thereby easing the problem of filtering.
On the other hand, we shall see that an increase in rate extends the bandwidth required for transmitting the sampled signal.
Accordingly an engineering compromise is called for.
An interesting special case is the sampling of a sinusoidal signal having the frequency fm.
Here, all the signal power is concentrated precisely at the cutoff frequency of the low-pass filter, and there is consequently some ambiguity about whether the signal frequency is inside or outside the filter passband.
To remove this ambiguity, we require that fs, >2fm, rather than that fs, >=2fm.
To see that this condition is necessary, assume that fs = 2fm, but that an initial sample is taken at the moment the sinusoid passes through zero.
Then all successive samples will also be zero. This situation is avoided by requiring fs>2fm.
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