The document discusses sampling theory and its importance. It explains that a continuous time signal can be reconstructed if it is sampled at a rate at least twice its highest frequency component. This is known as the sampling theorem or Nyquist rate. The Nyquist rate is the minimum sampling frequency required to perfectly reconstruct the original signal. The document provides examples of signals sampled at different rates, and explains the problems caused by under sampling. It also discusses applications of sampling theory in digital voice transmission.
The document discusses sampling theory and its importance. It explains that a continuous time signal can be reconstructed if it is sampled at a rate at least twice its highest frequency component. This is known as the sampling theorem or Nyquist rate. The Nyquist rate is the minimum sampling frequency required to perfectly reconstruct the original signal. The document provides examples of signals sampled at different rates, and explains the problems caused by under sampling. It also discusses applications of sampling theory in digital voice transmission.
Original Description:
It gives elementary idea about the A/D conversion and sampling process.
The document discusses sampling theory and its importance. It explains that a continuous time signal can be reconstructed if it is sampled at a rate at least twice its highest frequency component. This is known as the sampling theorem or Nyquist rate. The Nyquist rate is the minimum sampling frequency required to perfectly reconstruct the original signal. The document provides examples of signals sampled at different rates, and explains the problems caused by under sampling. It also discusses applications of sampling theory in digital voice transmission.
The document discusses sampling theory and its importance. It explains that a continuous time signal can be reconstructed if it is sampled at a rate at least twice its highest frequency component. This is known as the sampling theorem or Nyquist rate. The Nyquist rate is the minimum sampling frequency required to perfectly reconstruct the original signal. The document provides examples of signals sampled at different rates, and explains the problems caused by under sampling. It also discusses applications of sampling theory in digital voice transmission.
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Sampling Theory
and its importance
BHAVIN V KAKANI IT-NU Brief review : Time Domain signal Present a recurring phenomena as amplitude vs. time Sine Wave Sine Wave -1.5 -1 -0.5 0 0.5 1 1.5 0 1 2 3 4 5 6 Time A m p l i t u d e Frequency Domain signal Present recurring phenomena as amplitude vs. frequency Same sine wave looks like Frequency Multiple waves Multiple waves Both domains Voice in the Time Domain Voice in the Frequency Domain Sampling Theorem Sampling Theorem: A bandlimited signal can be reconstructed exactly if it is sampled at a rate atleast twice the maximum frequency component in it. OR A continuous-time signal may be completely represented in its samples and recovered back if the sampling frequency is fs2fm. Where fs is the sampling frequency and fm is the maximum frequency present in the signal. OR A band-limited signal of finite energy, which has no frequency component higher than fm Hz, is completely described by its sample values at uniform intervals less than or equal to 1 2 second apart. Figure 1 shows a signal g(t) that is bandlimited. Poor Sampling -1.5 -1 -0.5 0 0.5 1 1.5 0 2 4 6 8 10 12 Sampling Frequency = 1/2 X Wave Frequency Even Worse -1.5 -1 -0.5 0 0.5 1 1.5 0 2 4 6 8 10 12 Sampling Frequency = 1/3 X Wave Frequency Higher Sampling Frequency -1.5 -1 -0.5 0 0.5 1 1.5 0 2 4 6 8 10 12 Sampling Frequency = 2/3 Wave Frequency Getting Better -1.5 -1 -0.5 0 0.5 1 1.5 0 2 4 6 8 10 12 Sampling Frequency = Wave Frequency Good Sampling -1.5 -1 -0.5 0 0.5 1 1.5 0 2 4 6 8 10 12 Sampling Frequency = 2 X Wave Frequency Nyquist rate and Nyquist Interval When the sampling rate becomes exactly equal to 2fm samples per second, then it is called Nyquist rate. Nyquist rate is also called minimum sampling rate. It is given by fs = 2fm Hz
Similarly, Maximum sampling interval is called Nyquist interval. It is given by
Nyquist interval Ts = 1/2fm seconds Half the Nyquist Frequency -1.5 -1 -0.5 0 0.5 1 1.5 0 5 10 15 20 25 Nyquist Frequency -1.5 -1 -0.5 0 0.5 1 1.5 0 2 4 6 8 10 12 Digitizing Digital Voice Telephone Transmission Voice data for telephony purposes is limited to frequencies less than 4,000 Hz. According to Nyquist, it would take 8,000 samples (2 times 4,000) to capture a 4,000 Hz signal perfectly. Generally, one byte is recorded per sample (256 levels). One byte is eight bits of binary data. (8 bits * 8,000 samples per second = 64K bps) over a circuit. Problems Related to the topics: 1. What is the effect of under sampling? How it can be removed? 2. State and prove sampling theorem in time domain. 3. What is Nyquist rate and Nyquist interval? 4. A band limited signal x(t) is sampled by a train of rectangular pulses of width and period T. 1. Find an expression for the sampled signal. 2. Sketch the spectrum of the sampled signal. 5. What is anti-aliasing filter? 6. Define the sampling process and explain its necessity in communication system. 7. What do you understand by the word bandlimited? 8. What is Guard band? 9. A continuous-time signal is given below: X(t) = 8 cos 200t Determine I. Minimum sampling rate II. If the sampling frequency is 400Hz. What is the discrete time signal x[n]/x[nTs] obtained after sampling. 10. Determine the Nyquist rate for a continuous time signal X(t) = 6cos 50t + 20sin300 t -10cos100 t