SamplingBy

Muhammad Uzair Rasheed

MS Nuclear Power EngineeringKarachi Institute of Power

Engineering, Karachi

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.

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)

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

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

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.

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 thatx[n] = x(n·Ts)

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.

How sampling is done?(..)• For Input signal

• Known as discrete time frequency, Normalized continuous frequency.

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

Example(.)

Example(..)

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.

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

1 201

Impulse Samplingwith increasing sampling time T

Sampled waveform

0

1 201

Sampled waveform

0

1 201

Sampled waveform

0

1 201

Sampled waveform

01 201

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

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)

dejFkTf kTj)(2

1)(

• After some manipulations

n

T

T

kTj

n

T

njF

TDTFT

deT

njF

T

TkTf

21

21

2)(

1

Sampling Effects: Frequency Domain

Xc(j)

N-N

XS(j)

N-N S-S 2S-2S

S-S 2S-2S

XS(j)

S > 2 N

S < 2 N (aliasing)

Fourier Transform of continuous function

Fourier Transform of sampled function

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.

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.

Applications

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.

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.

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.

Audio Signals Sampling rate Use

8,000 HzTelephone and encrypted walkie-talkie, wireless intercom and wireless microphone transmission; 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 HzWideband 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.

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 HzDigital 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.

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.

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).

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.

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.

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.

dejFkTf kTj)(2

1)(

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.

Thank You !