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Pulse Modulation
• Pulse modulation schemes aim at transferring a narrowband analog signal over an analog baseband channel as a two-level signal by modulating a pulse wave.
• Some pulse modulation schemes also allow the narrowband analog signal to be transferred as a digital signal (i.e. as a quantized discrete-time signal) with a fixed bit rate, which can be transferred over an underlying digital transmission system, for example some line code. These are not modulation schemes in the conventional sense since they are not channel coding schemes, but should be considered as source coding schemes, and in some cases analog-to-digital conversion techniques.
Types• Analog-over-analog methods:• Pulse-amplitude modulation (PAM)• Pulse-width modulation (PWM) AND Pulse-depth modulation
(PDM)• Pulse-position modulation (PPM)• Analog-over-digital methods:• Pulse-code modulation (PCM)
– Differential PCM (DPCM)– Adaptive DPCM (ADPCM)
• Delta modulation (DM or Δ-modulation)• Delta-sigma modulation (∑Δ)• Continuously variable slope delta modulation (CVSDM), also called
Adaptive-delta modulation (ADM)• Pulse-density modulation (PDM)
Sampling Theory
Sampling process:
• In signal processing, sampling is the reduction of a continuous signal to a discrete signal.
• A sample refers to a value or set of values at a point in time and/or space.
• A theoretical ideal sampler produces samples equivalent to the instantaneous value of the continuous signal at the desired points.
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Periodic (Uniform) Sampling• Sampling is a continuous to discrete-time conversion
• Most common sampling is periodic
• T is the sampling period in second• fs = 1/T is the sampling frequency in Hz• Sampling frequency in radian-per-second s=2fs rad/sec• Use [.] for discrete-time and (.) for continuous time signals• This is the ideal case not the practical but close enough
– In practice it is implement with an analog-to-digital converters– We get digital signals that are quantized in amplitude and time
nnTxnx c
-3 -2 2 3 4-1 10
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Periodic Sampling• Sampling is, in general, not reversible• Given a sampled signal one could fit infinite continuous signals through the samples
0-1
20 40 60 80 100
-0.5
0
0.5
1
• Fundamental issue in digital signal processing– If we loose information during sampling we cannot recover it
• Under certain conditions an analog signal can be sampled without loss so that it can be reconstructed perfectly
Sampling rate
• The sampling rate, sample rate, or sampling frequency (fs) defines the number of samples per unit of time (usually seconds) taken from a continuous signal to make a discrete signal.
sampling period
• The sampling period is the time difference between two consecutive samples.
• It is the inverse of the sampling frequency. • For example: if the sampling frequency is
44100 Hz, the sampling period is 1/44100 = 2.2675736961451248e-05 seconds: the samples are spaced approximately 23 microseconds apart.
Instantaneous Sampling
Instantaneous sampling
• Consider an arbitrary signal g (t) of finite energy, which is specified for all time. Suppose that we sampled the signal g (t) instantaneously and at a uniform rate, once every Ts seconds. Consequently, we obtain an infinite sequence of samples spaced Ts seconds a part. W e refer to Ts as t he sampling period, and to its reciprocal fs = 1 /Ts as the sampling rate. This idle form of sampling called instantaneous sampling
Sampling TheoremSampling of
Band-Limited Signals
Band-Limited Signals
Yc(j)
Band-Limited
Band-Unlimited
Xc(j)
NN
1
Sampling of Band-Limited Signals
Band-Limited
TkjjX
TjX s
kscs
2 ),(
1)(
TkjjX
TjX s
kscs
2 ),(
1)(
Xc(j)
NN
1
ss 2s 3s2s3s
S(j)2/T
4s4s 2s 6s2s6s
S(j)2/T
Sampling withHigher Frequency
Sampling withLower Frequency
Nyquist–Shannon sampling theorem
• Sampling is the process of converting a signal (for example, a function of continuous time or space) into a numeric sequence (a function of discrete time or space).
• Shannon's version of the theorem states:If a function x(t) contains no frequencies higher
than B hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.
• In other words, a bandlimited function can be perfectly reconstructed from a countable sequence of samples if the bandlimit, B, is no greater than ½ the sampling rate (samples per second).
• The theorem also leads to a formula for reconstruction of the original function from its samples.
• When the bandlimit is too high (or there is no bandlimit), the reconstruction exhibits imperfections known as aliasing.
• The Poisson summation formula provides a graphic understanding of aliasing and an alternative derivation of the theorem, using the perspective of the function's Fourier transform.
Nyquist Theory
x(t) must contain no sinusoidal component at exactly frequency B, or that B must be strictly less than ½ the sample rate.
Recoverability
Band-Limited
TkjjX
TjX s
kscs
2 ),(
1)(
TkjjX
TjX s
kscs
2 ),(
1)(
Xc(j)
NN
1
ss 2s 3s2s3s
S(j)2/T
4s4s 2s 6s2s6s
S(j)2/T
Sampling withHigher Frequency
Sampling withLower Frequency
s > 2N s > 2N
s < 2N s < 2N
Case 1: s > 2N TkjjX
TjX s
kscs
2 ),(
1)(
TkjjX
TjX s
kscs
2 ),(
1)(
Xc(j)
NN
1
ss 2s 3s2s3s
S(j)2/T
1/T
ss 2s 3s2s3s
Xs(j)
Case 1: s > 2N TkjjX
TjX s
kscs
2 ),(
1)(
TkjjX
TjX s
kscs
2 ),(
1)(
Xc(j)
NN
1
ss 2s 3s2s3s
S(j)2/T
1/T
ss 2s 3s2s3s
Xs(j)
Passing Xs(j) through a low-pass filter with cutoff frequency N < c< s N , the original signal can be recovered.
Passing Xs(j) through a low-pass filter with cutoff frequency N < c< s N , the original signal can be recovered.
Xs(j) is a periodic function with period s.
Xs(j) is a periodic function with period s.
Case 2: s < 2N T
kjjXT
jX sk
scs
2 ),(
1)(
TkjjX
TjX s
kscs
2 ),(
1)(
Xc(j)
NN
1
1/T
2s2s 4s 6s4s6s
S(j)2/T
2s2s 4s 6s4s6s
Xs(j)
Case 2: s < 2N T
kjjXT
jX sk
scs
2 ),(
1)(
TkjjX
TjX s
kscs
2 ),(
1)(
Xc(j)
NN
1
1/T
2s2s 4s 6s4s6s
S(j)2/T
2s2s 4s 6s4s6s
Xs(j)Aliasi
ngAliasi
ng
No way to recover the original signal.No way to recover the original signal.
Xs(j) is a periodic function with period s.
Xs(j) is a periodic function with period s.
Nequist Rate
Xc(j)
NN
1Band-Limited
Nequist frequency (N) The highest frequency of a band-limited signal
Nequist rate = 2N
Nequist Sampling Theorem
Xc(j)
NN
1Band-Limited
s > 2N
s < 2N
Recoverable
Aliasing
Aliasing
• The Poisson summation formula shows that the samples, x(nT), of function x(t) are sufficient to create a periodic summation of function X(f).
• If the Nyquist criterion is not satisfied, adjacent copies overlap, and it is not possible in general to discern an unambiguous X(f). Any frequency component above fs/2 is indistinguishable from a lower-frequency component, called an alias, associated with one of the copies. In such cases, the reconstruction technique described below produces the alias, rather than the original component.
The samples of several different sine waves can be identical, when at least one of them is at a frequency above half the sample rate.
Mathematical Problems will be provided later.
©2000, John Wiley & Sons, Inc.Haykin/Communication Systems, 4th Ed
Figure 3.1The sampling process. (a) Analog signal. (b)
Instantaneously sampled version of the analog signal.
©2000, John Wiley & Sons, Inc.Haykin/Communication Systems, 4th Ed
Figure 3.2(a) Spectrum of a strictly band-limited signal
g(t). (b) Spectrum of the sampled version of g(t) for a sampling period Ts = 1/2 W.
©2000, John Wiley & Sons, Inc.Haykin/Communication Systems, 4th Ed
Figure 3.5Flat-top samples, representing an analog
signal.
©2000, John Wiley & Sons, Inc.Haykin/Communication Systems, 4th Ed
Figure 3.8Illustrating two
different forms of pulse-time modulation
for the case of a sinusoidal modulating
wave. (a) Modulating wave.
(b) Pulse carrier. (c) PDM wave. (d) PPM wave.
©2000, John Wiley & Sons, Inc.Haykin/Communication Systems, 4th Ed
Figure 3.10Two types of quantization: (a) midtread and (b) midrise.
©2000, John Wiley & Sons, Inc.Haykin/Communication Systems, 4th Ed
Figure 3.11Illustration of the quantization process. (Adapted from
Bennett, 1948, with permission of AT&T.)
