# What is sampling , sampling period (Ts) and sampling rate (fs), Guard band, Sampling Theorem

By   November 22, 2019

Guard band, Sampling Theorem , What is sampling , sampling period (Ts) and sampling rate (fs) :-
Q.1.     What is sampling ?
Ans.    The process of converting an analog signal into a discrete signal (or) making an analog (or) continuous signal to occur at a particular interval of time is known as sampling.
Q.2.     What do you mean by sampling period (Ts) and sampling rate (fs) ?
Ans.    During sampling, the time elapsed between sample to sample (or) time taken by the next sample occur is known as sampling period. It is denoted as Ts.
Sampling rate is reciprocal of sampling period. It is denoted as fs = 1/Ts.
DIAGRAM
FIGURE 3.37
Q.3.     What is Guard band ?
Ans.    When the sampling rate is chosen much higher than the Nyquist rate then a small space occur, between the samples. This space is said to be Guard band. This is the desired one for sampling the signals.
DIAGRAM
FIGURE 3.38
Q.4. What is Sampling Theorem?
Ans.    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 , i.e., 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 at the receiver end.
Q.5.     What is Natural Sampling?
Ans.    It is similar to that of Instantaneous Sampling.
Let us consider a baseband signal m(t) which is to be sampled. To sample m(t), a sampling function s(t) consisting of a train of pulses having duration T and separated by the sampling time Ts is given to one of the input of a product modulator and m(t) to other input. Hence, the output consists of pulses of duration T but with a varying amplitude. This is natural sampling.
DIAGRAM
FIGURE 3.39
Q.6.     What do you mean by aperture effect?
Ans.    During flat top sampling, to convert varying amplitudes of pulses to flat top pulses we use a sine function. Because of this, there would be decrease in the amplitude. This distortion is named as Aperture effect.
This may be eliminated by using an equalizer in cascade with the output low pass filter.
Q.7.     What is Pulse Amplitude Modulation (PAM)?
Ans.    The process in which amplitudes of regularly spaced rectangular pulses vary with the instantaneous sample values of a continuous message signal in a one to one fashion is known as pulse amplitude modulation.
Q.8.     What do you mean by synchronization in PAM systems?
Ans.    In general, most pulse systems require synchronization of the receiver to the transmitter. There are many ways to provide synchronization to the systems.
One of the method is a start-stop method of synchronization. This method involves transmitting some information, in addition to the message-bearing pulses, to serve as a time mark within each frame interval so that the gates in the receiver are made to open and close at the appropriate instants of time.
In some cases, the necessary time mark is established by transmitting a distinctive marker pulse per frame, whereas in some other cases, it is established by omitting a pulse in a particular time slot. One thing to be noted is that when markers are used, they should differ from the message-bearing pulses in some recognizable fashion.
In a PAM system, the marker pulse may be identified by making its amplitude exceed that of all possible message pulses.
Let us consider the transmission of PAM pluses with marker pulses. This has been shown in figure 3.40.
DIAGRAM
FIGURE 3.40
The PAM signal which is transmitted along with the marker pulses are detected at the receiver by giving these received pulses to a slicer.
The slicer is considered with a slicing level that is just in excess of the maximum amplitude of the message pulses, so that these pulses produces zero output.
An ideal slicer has the property that its output is zero whenever its input is below its slicing level and is constant whenever the input exceeds this level.
Q.9.     What do you mean by Pulse-Time Modulation?
Ans.    The modulation technique in which the time (or) duration of the pulses is varied in accordance with the amplitude of the message signal keeping the amplitude of the pluses constant is referred to as pulse-time modulation.
Q.10.   What are the different types of PTM systems?
Ans. There are two kinds of Pulse-time modulation schemes. They are:
(i)         Pulse duration (or) pulse width (or) pulse length modulation (PDM (or) PWM (or) PLN1).
(ii)        Pulse position modulation.
Q.11.   What is pulse duration modulation (PDM)?
Ans.    The method in which the samples of the message signal are used to vary the duration (or) width of the individual pluses. This is referred to as pulse duration modulation.
Q.12.   What is pulse position modulation (PPM)?
Ans.    In PPM, the position of a pulse relative to its unmodulated time of occurrence is varied in accordance with the message signal.
Q.13.   Describe the spectral representation of PDM and PPM waves.
Ans.    The spectral analysis of a PDM and PPM wave is complicated.
Let us consider the qualitative description of the spectra of PDM and PPM waves.
Let Ls denotes the time separation between the leading edges of duration modulated pluses obtained by natural sampling.
fm is sinusoidal modulation wave of frequency.
Thus the spectrum of a naturally sampled PDM wave consists of the following components:

• Sinusoidal components of frequencies equal to the integer multiplies of l/Ts, corresponding to spectral lines at , where n = 1, 2, 3,… These sinusoidal components as well as the DC components are contributed by the unmodulated pulse train which is regarded as the carrier of the PDM wave.
• A sinusoidal component of frequency fm, and in phase with modulating wave, corresponding to spectral lines at ±fm.
• Sinusoidal components of frequencies equal to pairs of side-frequencies centred around each spectral line of the unmodulated pulse train, expect the dc component.

It may be noted that these dc components represent the cross-modulation products between the sinusoidal modulation and sampling frequencies.
(i)         Unlike, PAM, noise is less, since in PWM, amplitude is held constant.
(ii)        Signal and noise separation is very easy.
(iii)       PWM communication does not require synchronization between transmitter and receiver.
(i)         In PWM, pulses are varying in width and therefore their power contents are variable. This requires that the transmitter must be able to handle the power contents of the pulse having maximum pulse width.
(ii)        Large bandwidth is required for the PWM communication as compared to PAM.
(i)         Like PWM, in PPM, amplitude is held constant thus less noise interference.
(ii)        Like PPM, signal and noise separation is very easy.
(iii)       Because of constant pulse widths and amplitudes, transmission power for each pulse is same.
(i)         Synchronization between transmitter and receiver is required.
(ii)        Large bandwidth is required as compared to PAM.
QUESTIONS

1. State and prove sampling theorem in time domain.
2. What is Nyquist rate and Nyquist interval?
3. A bandlimited signal x(t) is sampled by a train of rectangular pulses of width and period T.

(i)         Find an expression for the sampled signal.
(ii)        Determine the spectrum of the sampled signal and sketch it.

1. What is aliasing and how it is reduced?

PROBLEMS

1. Determine the Nyquist sampling rate and the Nyquist sampling interval for the following signals:

(a) sinc                                             (b) Sinc2
(c) sinc  + sinc         (d) sinc +3sinc2
(e) sinc

1. A signal g(t) bandlimited to BHz is sampling by a periodic pulse train pTs(t) made up of a rectangular pulse of with second (centred at the origin) repeating at the Nyquist rate (2B pulses per second). Show that the sampled signal  is given by

EQUATION
Show that the signal g(t) can recovered by passing (t) through an ideal low-pass filter of bandwidth BHz and a gain of 4.

1. A signal g(t) sinc2(is sampled (using uniformly spaced impulses) at a rate of: (i) 5Hz; (ii) 10 Hz; (iii) 20 Hz. Now, for each of the three cases:

(a)        Sketch the sampled signal.
(b)        Sketch the spectrum of the sampled signal.
(c)        Explain whether you can recover the signal g(t) from the sampled signal.

1. Signal g1(t) = 104rect (104t) and g2(t) = are applied at the inputs of ideal low-pass filters H1 = rect (000 and H2( = rest (000 (Figure 3.41). The outputs y1(t) and y2(t) of these filters are multiplied to obtain the signal y(t) = y1(t) y2(t). Find the Nyquist rate of y1(t), y2(t), and y(t).

(GATE Examniation-1998)
DIAGRAM
FIGURE 3.41
5.         A zero-order hold circuit (Figure 3.42) is often used to reconstruct a signal g(t) from its samples.
DIAGRAM
FIGURE 3.42
(a)        Find the unit impulse response of this circuit.
(b)        Find the transfer function H() and sketch
(c)        Show that when a sampled signal  is sampled at the input of this circuit, the output is a staircase approximation of g(t). The sampling interval is Ts.                                           (Pune University, 1998)
OBJECTIVE TYPE QUESTIONS

1. Fill up the Blanks
2. _________ is the process in which the analog signal is converted into a corresponding sequence of samples that the are uniformly spaced in time.
3. In the process of uniformly sampling, a signal in the time domain results in a periodic spectrum is the frequency domain with a period equal to the ___________.
4. The sampling rate (fs) of value 2W samples per second for a signal bandwidth of W Hz is often referred to as ______________.
5. Prior to sampling a ____________ is used to attenuate the high frequency components of the signal that lie outside the band of interest.
6. In ____________the amplitudes of regularly spaced rectangular pulses vary with the instantaneous sample values of a continuous message signal in a one-to-one fashion.
7. The figure of merit of a PPM system is proportional to the square of ____________.
8. PPM and PDM systems suffer from a ____________ similar to that experienced in FM systems.
9. The sampled wave in practical system consists of____________ and ____________ rather then impulses.
10. The term____________ refers to the signal from an adjacent channel spilling over into a desired time slot.
11. ____________ is the process which involves sampling of the lowpass in-phase and quadrature components of the bandpass signal in accordance with the sampling theorem.
12. In____________ the samples of the message signal are used to vary the duration of the individual pulses.
13. Pulse duration modulation is also known as____________(or) ____________.
14. In____________ the position of a pulse relative to its unmodulated time of occurrence is varied in accordance with the message signal.
15. The bandpass signal is represented as a combination ____________ and ____________components.
16. Multiple Choice Questions
17. The aperture effect in flat top pulses is reduced by using an

(a) Predictor                            (b) Integrator
(c) Equalizer                            (d) Compander

1. The narrow samples produced at the pulse demodulator output are distributed to appropriate low pass reconstruction filter by means of

(a) Commutator                                   (b) Multiplexer
(c) Decommutator                               (d) none of the above

1. Which of the following method is employed in telephony

(a) Time division multiplexing
(b) Frequency division multiplexing
(c) both (a) and (b)
(d) only (a).

1. If a time function contains no frequency components higher than W Hz, then the time function can be completely determined by specifying its ordinates at a series of points spaced every or less

(a) true                                                 (b) false

1. If a time function is composed df a band of frequencies displaced from zero with a bandwidth of W Hz with highest frequency f2, then minimum sampling rate is given by 2f2.

(a) true                                                 (b) false

1. If a signal height whose higest frequency is W Hz has been sampled at a rate 2 W samples/sec; and the samples are in the form of impulses whose area is proportional to the magnitude of the sample at that instant, the sampled signal may be reconstructed by passing the impulse train through an ideal low pass filter whose cut off frequencies is 2 W Hz.

(a) true                                                 (b) false

1. Pulse amplitude modulation is a process whereby

(a)        the position of the public is changed as a function of the sampled value
(b)        the width of the pulse is varied as a function of time
(c)        the height of a pulse is made proportional to the sampled value
(d)       none of these

1. Pulse width of modulation is a process whereby

(a)        the position of a pulse is changed as a function of the sampled value
(b)        the sampled value is first coded and then transmitted
(c)        the width of a pulse is varied as a function of the sampled value
(d)       none of these

1. The signal-to-noise (S/N) ratio appearing at the output of each channel in PAM is

(a)        three times the input S/N
(b)        one-and-half times the input S/N
(c)        twice the input SIN
(d)       none of these

1. Pulse width modulation requires more power than pulse amplitude modulation

(a) true                                                 (b) false.

1. In PCM a system, the quantization noise depends upon               (IETE, 1998)

(a) the number of quantization levels only
(b) the sampling rate only
(c) both the sampling rate and the number of quantization levels
(d) none of the above is correct

1. In a pulse position modulation system, the transmitted pulse have                                                        (IETE, 1999)

(a)        constant amplitudes but varying widths
(b)        constant amplitudes and constant widths
(c)        constant width but varying amplitude
(d)       none of the above

1. PAM signal can be demodulated by using        (IETE, 1998)

(a)        a low pass filter
(b)        a bandpass filter
(c)        a high pass filter
(d)       none of the above.

1. In pulse modulation, the number of samples required to ensure no loss of information is given by                                                        (IETE, 1998)

(a)        Nyquist theorem
(b)        Parsevals theorem
(c)        Fourier transform

1. Compared to transistor and FETs, the speed of switching in a Schottky diode is                                                        (IETE, 1998)

(a)        higher                                      (b) lower
(c)        same

1. Aperture effect occurs in communication due to

(a)        sampling at less than Nyquist rate
(b)        flat top sampling
(c)        finite bandwidth of transm-ission channel
(d)       short duration of samples

1. Which of the following require a synchronizing signal.

(a)        PPM                                        (c) PDM
(b)        PAM                                       (d) all of the above

1. PAM signals can be demodulated by using

(a) 1 LPF alone
(b) a schmitt trigger followed by a LPF
(c) a differentiator followed by a LPF
(d) a clipper circuit followed by a LPF

1. The Nyquist sampling rate for a signal band limited to 4 kHz is

(a) 4 kHz                                             (b) 8 kHz
(c) 2 kHz                                             (d) 16 kHz

1. The Nyquist rate of sampling for the signal x(t)=sinc (200t) + sinc2(200t) is

(a) 200                                                 (b) 400
(c) 300                                                 (d) 250

1. A message signal bandlimited to 5 kHz is sampled at the minimum rate as dictated by the sampling theorem. The number of quantization levels is 64. If the samples are encoded in binary form, the transmission rate is

(a) 60 Mbps                                         (b) 50 Mbps
(c) 32 Kbps                                         (d) 10 Kpbs