Brief and simple concepts of Sampling, Nyquist, Quantization and ADC

1. Sampling
By
Muhammad Uzair Rasheed
MS Nuclear Power Engineering
Karachi Institute of Power Engineering, Karac

2. Sampling
• The signals we use in the real world, such as our voices, are called
"analog" signals.
• To process these signals in computers, we need to convert the signals
to "digital" form.
• Analog signal is continuous in both time and amplitude, a digital
signal is discrete in both time and amplitude.

3. Sampling(.)
• To convert a signal from continuous time to discrete time, a process
called sampling is used. The value of the signal is measured at certain
intervals in time.
• Each measurement is referred to as a sample. (The analog signal is
also quantized in amplitude, but that process is ignored as it will be
explained by some other group)

4. Figure 0: Signal sampling representation.
The continuous signal is a green colored line
Discrete samples are indicated by the blue vertical lines.

5. Sampling(…)
• Sampling can be done for functions varying in space, time, or any
other dimension, and similar results are obtained in two or more
dimensions.
• The sampling frequency or sampling rate, fs, is the average number
of samples obtained in one second (samples per second), thus
fs= 1/T

6. Sampling(..)
• A sample is a value or set of values at a point in time and/or space.
• A sampler is a subsystem or operation that extracts samples from a
continuous signal.
• A theoretical ideal sampler produces samples equivalent to the
instantaneous value of the continuous signal at the desired points.

7. How sampling is done?
• First obtain signal values from the continuous signal at regular time-
intervals (Ts). Which is sampling time and its reciprocal is fs sampling
frequency
• The result of this process is just a sequence of numbers.
• Our discrete time signal will be denoted as x[n] where n is index.
• As sampling interval Ts is defined, sampling just extracts the signals
value at all integer multiples of Ts such that
x[n] = x(n·Ts)

8. How sampling is done?(.)
• At this point (after sampling), our signal is not yet completely digital
because the values x[n] can still take on any number from a
continuous range.
• So we use the terms discrete-time signal.
• Figure 1 illustrates the process of sampling a continuous sinusoidal.

9. How sampling is done?(..)
• For Input signal
• Known as discrete time frequency, Normalized continuous frequency.

10. Example: Sampling rate Comparisons
• Consider at sampling rates of 240 and 1000 samples per second

11. Example(.)

12. Example(..)

13. Sampling Tool
• In practice, the continuous signal is sampled using analog-to-digital
converter (ADC), a device with various physical limitations. This
results in deviations from the theoretically perfect reconstruction,
collectively referred to as distortion.

14. Impulse Sampling
Sampled waveform
0
1 201
Sampled waveform
0
1 201
Sampled waveform
0
1 201
Signal waveform
0
1 201
Impulse sampler
0

15. Impulse Sampling
with increasing sampling time T
Sampled waveform
0
1 201
Sampled waveform
0
1 201
Sampled waveform
0
1 201
Sampled waveform
0
1 201

16. Natural sampling
(Sampling with rectangular waveform)
Signal waveform
0
1 201 401 601 801 1001 1201 1401 1601 1801 2001
Natural sampler
0
1 201 401 601 801 1001 1201 1401 1601 1801 2001
Sampled waveform
0
1 201 401 601 801 1001 1201 1401 1601 1801 2001

17. Fourier Relationship
• The relation between continuous-time signal f(t) and a discrete-time
(sampled) signal f(kT)
• where T is the time between samples (T=1/fs)
∫
∞
∞−
= ωω
π
ω
dejFkTf kTj
)(
2
1
)(

18. • After some manipulations


















+=


















+=
∑
∫ ∑
∞
−∞=
−
−
∞
−∞=
n
T
T
kTj
n
T
n
jF
T
DTFT
de
T
n
jF
T
T
kTf
π
ω
ω
π
ω
π
π
π
ω
21
21
2
)(
1

19. Sampling Effects: Frequency Domain
Xc(jΩ)
Ω
ΩN-ΩN
XS(jΩ)
Ω
ΩN-ΩN ΩS-ΩS 2ΩS-2ΩS
Ω
ΩS-ΩS 2ΩS-2ΩS
XS(jΩ)
ΩS > 2 ΩN
ΩS < 2 ΩN (aliasing)
Fourier Transform of
continuous function
Fourier Transform of
sampled function

20. Under Sampling
• In Undersampling a band pass signal is sampled slower than
its Nyquist rate.
• When one undersamples a bandpass signal, the samples are
indistinguishable from the samples of a low-frequency samples of the
high-frequency signal.
• In such a way that the lowest-frequency alias satisfies the Nyquist
criterion, because the bandpass signal is still uniquely represented
and recoverable. Such undersampling is also known as bandpass
sampling, harmonic sampling, IF sampling, and direct IF to digital
conversion.

21. Oversampling
• In Oversampling a signal is sampled faster than its Nyquist rate.
• Oversampling is used in most modern analog-to-digital converters to
reduce the distortion or noise effects introduced by practical digital-
to-analog converters.

22. Applications

23. Audio sampling
• Digital audio uses pulse-code modulation and digital signals for sound
reproduction.
• Includes analog-to-digital conversion (ADC), digital-to-analog
conversion (DAC), storage, and transmission.
• The system is commonly referred to as digital is in fact a discrete-
time, discrete-level. The primary usefulness of a digital system is the
ability to store, retrieve and transmit signals without any loss of
quality.

24. PCM(Pulse Code Modulation)
• Pulse-code modulation (PCM) is a method used to digitally represent
sampled analog signals. It is the standard form of digital audio in
computers, Compact Discs, digital telephony and other digital audio
applications. In a PCM stream, the amplitude of the analog signal is
sampled regularly at uniform intervals, and each sample
is quantized to the nearest value within a range of digital steps.

25. Audio Sampling(.)
• Audio covers the entire 20–20,000 Hz range of human hearing,
• Recording music or many types of acoustic events, audio waveforms
are typically sampled at 44.1 kHz (CD Music), 48 kHz, 88.2 kHz, or
96 kHz.
• The approximately double-rate requirement is a consequence of
the Nyquist Criteria.
• Sampling rates higher than about 50 kHz to 60 kHz cannot supply
more usable information for human listeners. Early professional
audio equipment manufacturers choose sampling rates in the region
of 50 kHz for this reason.

26. Audio Signals
Sampling rate Use
8,000 Hz
Telephone and encrypted walkie-talkie, wireless intercom and wireless
microphonetransmission; adequate for human speech but
without sibilance; ess sounds like eff (/s/, /f/).
11,025 Hz
One quarter the sampling rate of audio CDs; used for lower-quality
PCM, MPEG audio and for audio analysis of subwoofer bandpasses.
16,000 Hz
Wideband frequency extension over
standard telephone narrowband 8,000 Hz. Used in most
modern VoIP and VVoIP communication products.[13]
32,000 Hz
miniDV digital video camcorder, video tapes with extra channels of
audio (e.g. DVCAM with 4 Channels of audio), DAT(LP mode),
Germany's Digitales Satellitenradio, NICAM digital audio, used alongside
analogue television sound in some countries. High-quality
digital wireless microphones.Suitable for digitizing FM radio.

27. Audio Signals(.)
50,000 Hz
First commercial digital audio recorders from the late 70s
from 3M and Soundstream.
50,400 Hz Sampling rate used by the Mitsubishi X-80 digital audio recorder.
176,400 Hz
Sampling rate used by HDCD recorders and other professional
applications for CD production.
192,000 Hz
DVD-Audio, some LPCM DVD tracks, BD-ROM (Blu-ray Disc) audio tracks,
and HD DVD (High-Definition DVD) audio tracks, High-Definition audio
recording devices and audio editing software. This sampling frequency is
four times the 48 kHz standard commonly used with audio on
professional video equipment.
352,800 Hz
Digital eXtreme Definition, used for recording and editing Super Audio
CDs, as 1-bit DSD is not suited for editing. Eight times the frequency of
44.1 kHz.

28. Video Sampling
• Standard-definition television (SDTV) uses either 720 by 480 pixels or
704 by 576 pixels for the visible picture area.
• High-definition-television (HDTV)uses 720p , and 1080p.
• In digital video, the sampling rate is defined as the frame rate.
• The image sampling frequency is the repetition rate of the image
sensor integration period.

29. Video Sampling(.)
• When analog video is converted to digital video, a sampling process
occurs, here the pixel frequency, corresponds to a spatial sampling
rate. A common pixel sampling rate is:
13.5 MHz – CCIR 601, D1 video
• Video sensors operate in the megahertz range (from ~3 MHz for low
quality composite video in early games consoles, to 250 MHz or more
for the highest-resolution VGA output).

30. Speech sampling
• Speech signals, i.e., signals intended to carry only human speech, can
usually be sampled at a much lower rate.
• Mostly almost all of the energy is contained in the 5Hz-4 kHz range,
allowing a sampling rate of 8 kHz.
• This is the sampling rate used by nearly all telephony systems, which
use the G.711 sampling and quantization specifications.

31. References
• Martin H. Weik (1996). Communications Standard Dictionary.
Springer.
• Rao, R. Signals and Systems. Prentice-Hall Of India Pvt. Limited.
• C. E. Shannon, "Communication in the presence of noise", Proc.
Institute of Radio Engineers, vol. 37, no.1, pp. 10–21, Jan.

32. Question / Answers
• What is an ideal Sampler?
• A theoretical ideal sampler produces samples equivalent to the instantaneous value of the
continuous signal at the desired points.
• Write the relation between continuous-time signal f(t) and a discrete-
time (sampled) signal f(kT)?
• What is Distortion?
• In DAC , conversion from digital back to analog, the deviations from the
theoretically perfect reconstruction, collectively referred to as distortion.
∫
∞
∞−
= ωω
π
ω
dejFkTf kTj
)(
2
1
)(

33. Question/ Answers
• What is PCM(Pulse Code Modulation)?
• A method used to digitally represent sampled analog signals. It’s the standard
form of digital audio in PC’s, CD’s & digital telephony etc. In a PCM stream,
the amplitude of the analog signal is sampled regularly at uniform intervals,
and each sample is quantized to the nearest value within a range of digital
steps.
• Difference Between Undersampling and Oversampling?
• In Undersampling a band pass signal is sampled slower than its Nyquist rate,
while in Oversampling a signal is sampled faster than its Nyquist rate.

34. Thank You !