Digital Communications Tutorial Cognitive Radio Communications @ Virginia Tech NSF Research Experiences for Undergraduates (REU) Site Ratchaneekorn (Kay) Thamvichai [email protected]
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Digital Communications Tutorial Cognitive Radio Communications @ Virginia Tech NSF Research Experiences for Undergraduates (REU) Site Ratchaneekorn (Kay)
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Slide 1
Digital Communications Tutorial Cognitive Radio Communications
@ Virginia Tech NSF Research Experiences for Undergraduates (REU)
Site Ratchaneekorn (Kay) Thamvichai [email protected]
Analog vs. Digital Transmitted bits can be detected and
regenerated, so noise does not propagate additively. More signal
processing techniques are available to improve system performance:
source coding, channel (error-correction) coding, equalization,
encryption, filtering, Digital ICs are inexpensive to manufacture
Digital communications permits integration of voice, video, and
data on a single system (ISDN) Implementation of various algorithms
can be done by software instead of hardware Security is easier to
implement.
Slide 4
Simple Digital Communication System Diagram
Slide 5
Digital communications system block diagram
Slide 6
Fourier Transform F() is the continuous-time Fourier transform
of f(t). The Fourier transformation F() is the frequency domain
representation of the original function f(t). It describes which
frequencies are present in the original function.
Slide 7
Example 1: Ex A: Find the Fourier Transform of x(t) = (t) Ex B:
Find the Fourier Transform of x(t) = 0.5cos(500t) t=0 x(t)
Slide 8
Ex C: Find the Fourier Transform of f(t) = rect(t/)
Slide 9
sinc function sinc(x) = sin(x) x even function zero crossings
at Amplitude decreases proportionally to 1/x 0 1 x sinc(x)
Linear Time-Invariant (LTI) system Convolution: y(t) =
x(t)*h(t) Its Fourier Transform: Y() = X()H() where H() is a
frequency response or a transfer function of a system h(t).
h(t)
Slide 12
Ideal filters A filter is used to eliminate unwanted parts of
the frequency spectrum of a signal. A filter is LTI system with an
impulse response h(t). The output y(t) of a filter can be founded
in time domain using a convolution. However, it is easier to do it
in a frequency domain: Y() = X()H()
Slide 13
Low Pass Filter with a cutoff frequency c High Pass Filter
Slide 14
Example 2: Given x(t) = cos(500t)cos(1000t), find an impulse
response h(t) of a low-pass filter that passes the low frequency
component of the signal. x(t) y(t) = low freq. component of x(t)
Low-pass filter h(t)
Sampling Continuous-Time signals Sampling generating of an
ordered number of sequence by taking values of f(t) as specified
instants of time i.e. f(t 1 ), f(t 2 ), f(t 3 ), where t m are
instants at which sampling occurs. Sampling operation is
implemented in hardware by an analog-to-digital converter (ADC)
electronic device used to sample physical voltage signals. In most
cases, continuous-time signals are sampled at equal increments of
time. The sample increment, called sample period, is usually
denoted as T s.
Slide 18
Impulse sampling Define the continuous time impulse train as:
p(t) is an infinite train of continuous time impulse functions,
spaced T s seconds apart.
Slide 19
Let x(t) be a continuous time signal we wish to sample. We will
model sampling as multiplying a signal x(t) by p(t).
Slide 20
Sampling Theorem let P() be a Fourier Transform of p(t), X() be
a Fourier Transform of x(t), X s () be a Fourier Transform of x s
(t), Since x s (t) = x(t)p(t) by a multiplication property (Fourier
Transform),
Slide 21
where C k are the Fourier Series coefficients of the periodic
signal. 21
Slide 22
We see that an impulse train in time, p(t), has a Fourier
Transform that is an impulse train in frequency, P(). The spacing
between impulses in time is T s, and the spacing between impulses
in frequency is 0 = 2/T s. Note: If we increase the spacing in time
between impulses, this will decrease the spacing between impulses
in frequency, and vice versa.
Slide 23
Spectrum of a sampled signal replicated scaled versions of X(),
spaced every 0 apart in frequency
Slide 24
Time-domain Frequency-domain 0 = 2/T s
Slide 25
If c < c, ALIASING (overlap area) occurs If c c, Note: if 0
- c c or 0 2 c, there is no aliasing
Slide 26
Sampling Theorem Let x(t) be a band-limited signal with X() = 0
for || > c. Then x(t) is uniquely determined by its samples x(nT
s ), n = 0, 1, 2, if 0 2 c where 0 = 2/T s. This is how to choose a
sampling frequency (f s = 1/T s ) or period (T s ) such that an
original continuous-time signal x(t) can be recovered from a
sampled version x s (t). => a sampling rate ( 0 ) MUST be at
least twice the highest frequency ( c ) of a signal to avoid
aliasing problem.
Slide 27
To recover x(t) from its sampled version x s (t), we use a low
pass filter (reconstruction filter) to recover the center island of
X s ():
Slide 28
Ex: Given a signal x(t) with Fourier Transform with cutoff
frequency c as shown: Given three different pulse trains with
periods Draw the sampled spectrum in each case. Which case(s)
experiences aliasing?
Slide 29
Slide 30
Aliasing Phenomenon Sampling theorem: the signal is strictly
band-limited ( c ). However, in practice, no information-bearing
signal is strictly band-limited. Aliasing is the phenomenon of a
high-frequency component in the spectrum of the signal seemingly
taking on the identify of a lower frequency in the spectrum of its
sampled version. To prevent the effects of aliasing in practice
Prior to sampling : a low-pass anti-alias filter is used to
attenuate those high-frequency components of a message signal that
are not essential to the information being conveyed by the signal.
The filtered signal is sampled at a rate slightly higher than the
Nyquist rate.
Slide 31
Example: Why 44.1 kHz for Audio CDs? Sound is audible in 20 Hz
to 20 kHz range: f max = 20 kHz and the Nyquist rate 2f max = 40
kHz What is the extra 10% of the bandwidth used? Rolloff from
passband to stopband in the magnitude response of the anti-aliasing
filter. Okay, 44 kHz makes sense. Why 44.1 kHz? At the time the
choice was made, only recorders capable of storing such high rates
were VCRs. NTSC: 60-Hz video (30 frames/s) - 490 lines per frame or
245 lines per field, 3 audio samples per line the sampling rate is
60 X 245 X 3 = 44.1 KHz
Pulse-Amplitude Modulation (PAM) The amplitude of regularly
spaced pulses are varied in proportion to the corresponding sample
values of a continuous message signal. Two operations involved in
the generation of the PAM signal Instantaneous sampling of the
message signal m(t) every T s seconds, Lengthening the duration of
each sample, so that it occupies some finite value T.
Slide 34
Sample-and-Hold Filter : Analysis The PAM signal is The h(t) is
a standard rectangular pulse of unit amplitude and duration The
instantaneously sampled version of m(t) is
Slide 35
Slide 36
To modify m (t) so as to assume the same form as the PAM
signal: The PAM signal s(t) is mathematically equivalent to the
convolution of m (t), the instantaneously sampled version of m(t),
and the pulse h(t). Its Fourier Transform:
Slide 37
Slide 38
One benefit of PAM It enables the simultaneous transmission of
multiple signals using time-division multiplexing (TDM). User 1
User 2
Slide 39
39 Quantization Process Amplitude quantization: The process of
transforming the sample amplitude m(nT s ) of a baseband signal
m(t) at time t=nT s into a discrete amplitude v(nT s ) taken from a
finite set of possible levels. It will be represented by binary
number(s)
Baseband Transmission of Digital Data The transmission of
digital data over a physical communication channel is limited by
two unavoidable factors 1.Intersymbol interference 2.Channel
noise
Slide 42
42
Slide 43
43 The level-encoded signal and the discrete PAM signal are The
transmitted signal is The channel output is The output from the
receive-filter is
Slide 44
44 The InterSymbol Interference (ISI) Problem We may express
the receive-filter output as the modified PAM signal where After
sampling:
Slide 45
45 ISI (cont.) Define where E is the transmitted signal energy
/ bit (symbol). What we desire is However, from Residual
phenomenon, intersymbol interference (ISI)
Slide 46
Pulse-shaping Given the channel transfer function, determine
the transmit-pulse spectrum and receive-filter transfer function so
as to satisfy two basic requirements: 1.Intersymbol interference
(ISI) is reduced to zero. 2.Transmission bandwidth is
conserved.
Slide 47
47 The Nyquist Channel The optimum solution for zero ISI at the
minimum transmission bandwidth possible in a noise-free environment
For zero ISI, it is necessary for the overall pulse shape p(t) and
the inverse Fourier transform of the pulse spectrum P(f) to satisfy
the condition
Slide 48
48 The overall pulse spectrum is defined by the optimum brick-
wall function: The brick-wall spectrum defines B 0 as the minimum
transmission bandwidth for zero intersymbol interference. The
optimum pulse shape is the impulse response of an ideal low-pass
channel with an amplitude response P opt (f) in the passband and a
bandwidth B 0
Slide 49
49
Slide 50
Symbol 1 Symbol 2 Symbol 3
Slide 51
51 Two difficulties that make its use for a PAM system
impractical: 1.The system requires that the spectrum P(f) be flat
from B 0 to B 0, and zero elsewhere 2.The time function p(t)
decreases as 1/|t| for large |t|, resulting in a slow rate of
decay
Slide 52
52 Raised-Cosine Pulse Spectrum To ensure physical
realizability of the overall pulse spectrum P(f), the modified P(f)
decreases toward zero gradually rather than abruptly 1.Flat
portion, which occupies the frequency band 0|f| f 1 for some
parameter f 1 to be defined 2.Roll-off portion, which occupies the
frequency band f 1 |f| 2B 0 -f 1
Slide 53
53 The roll-off factor: Time-domain of the overall channel The
amount of intersymbol interference resulting from a timing error t
decreases as the roll-off factor is increased form zero to
unity.
Slide 54
54 Frequency domain P(f) Time domain p(t)
Slide 55
55 Transmission-Bandwidth Requirement The transmission
bandwidth required by using the raised-cosine pulse spectrum is
Excess channel The transmission bandwidth requirement of the
raised-cosine spectrum exceeds that of the optimum Nyquist channel
by the amount 1.When the roll-off factor is zero, the excess BW is
reduced to zero 2.When the roll-off factor is unity, the excess BW
is increased to B 0.
Slide 56
56 Summary (ISI) The intersymbol interference problem, which
arises due to imperfections in the frequency response of the
channel ISI refers to the effect on that pulse due to cross-talk or
spillover from all other signal pulses in the data stream applied
to the channel input A corrective measure widely used in practice
is to shape the overall pulse spectrum of the baseband system,
starting from the source of the message signal all the way to the
receiver. ISI is a signal-dependent phenomenon, it therefore
disappears when the information-bearing signal is switched off.
Noise is always there, regardless of whether there is data
transmission or not. Another corrective measure for dealing with
the ISI: channel equalization.
58 Digital band-pass modulation techniques Baseband
Communication: Signals are transmitted without any shift in the
range of frequency of the signal. Band-pass Communication: Uses
modulation to shift the frequency spectrum of a (carrier)
sinusoidal signal. Usually, one of the basic parameters (amplitude,
frequency, or phase) of the carrier signal is varied in proportion
to the baseband signal (information-bearing data stream). Why
modulate signals? Convert signals to a form that is suitable for
transmission Sharing the frequency band with other stations Three
basic modulation schemes: Amplitude-shift keying (ASK) Phase-shift
keying (PSK) Frequency-shift keying (FSK)
Slide 59
Given a binary source The modulation process involves switching
or keying the amplitude, phase, or frequency of a sinusoidal
carrier wave between a pair of possible values in accordance with
symbol (bit) 0 and 1. Examples of a band-pass process 1.Binary
amplitude shift-keying (BASK) The carrier amplitude is keyed
between the two possible values used to represent symbols 0 and 1
2.Binary phase-shift keying (BPSK) The carrier phase is keyed
between the two possible values used to represent symbols 0 and 1.
3.Binary frequency-shift keying (BFSK) The carrier frequency is
keyed between the two possible values used to represent symbols 0
and 1.
Slide 60
60
Slide 61
61 In digital comm., the usual practice is to assume that the
carrier c(t) has unit energy measured over one symbol (bit)
duration (T b ). where Decreasing the bit duration T b has the
effect of increasing the transmission bandwidth requirement of a
binary modulated wave. (Fourier Transform property).
Slide 62
62 Band-Pass Assumption The spectrum of a digital modulated
wave s(t) is centered on the carrier frequency f c where b(t) is an
incoming binary stream with bandwidth W. Assumption: f c >>
BW, There will be no spectral overlap in the generation of s(t )
The transmitted signal energy per bit can be approximated as:
Slide 63
Binary Amplitude-Shift Keying (BASK) The ON-OFF signaling
variety The average transmitted signal energy is (the two binary
symbols must be equi-probable)
Slide 64
64
Slide 65
65 f c = 8 Hz, T b = 1s
Slide 66
66 f c = 8 Hz, T b = 0.5 s
Slide 67
67 From figures: The spectrum of the BASK signal contains a
line component at f=f c When the carrier is fixed and the bit
duration is halved, the width of the main lobe of the sinc function
defining the envelope of the BASK spectrum is doubled, which, in
turn, means that the transmission bandwidth of the BASK signal is
doubled. T b halved W is doubled The transmission bandwidth of
BASK, measured in terms of the width of the main lobe of its
spectrum, is equal to 2/T b, where T b is the bit duration.
Slide 68
68 Phase-Shift Keying Binary Phase-Shift Keying (BPSK) The pair
of signals used to represent symbols 1 and 0, An antipodal signals
A pair of sinusoidal wave, which differ only in a relative
phase-shift of radians. Note: The transmitted energy per bit, E b,
is constant. Equivalently, the average transmitted power is
constant.
Slide 69
Signal Space diagram of BPSK
Slide 70
70
Slide 71
71 f c = 8 Hz, T b = 1s
Slide 72
72 f c = 8 Hz, T b = 0.5 s
Slide 73
73 From figures: BASK and BPSK signals occupy the same
transmission bandwidth (2/T b ), which defines the width of the
main lobe of the sinc-shaped power spectra. The BASK spectrum
includes a carrier component, whereas this component is absent from
the BPSK spectrum.
Slide 74
74 Quadriphase-Shift Keying (QPSK) An important goal of digital
communication is the efficient utilization of channel bandwidth. In
QPSK, the phase of the sinusoidal carrier takes on one of the four
equally spaced values, such as /4, 3/4, 5/4, and 7/4 Each one of
the four equally spaced phase values corresponds to a unique symbol
which is a pair of bits (00, 01, 10, 11). Symbol duration
Slide 75
75 1.In reality, the QPSK signal consists of the sum of two
BPSK signals. 2.One BPSK signal, represented by the first term
defined the product of modulating a binary wave by the sinusoidal
carrier 3.The second binary wave
Slide 76
76
Slide 77
Signal Space diagram of QPSK
Slide 78
78
Slide 79
QPSK Transmitter
Slide 80
80 QPSK Receiver
Slide 81
81 f c = 8 Hz, T b = 1s BW = 1/T b
Slide 82
82 f c = 8 Hz, T b = 0.5 s BW = 1/T b
Slide 83
83 Frequency-Shift Keying Binary Frequency-Shift Keying (BFSK)
Each symbols are distinguished from each other by transmitting one
of two sinusoidal waves that differ in frequency by a fixed amount:
Sundes BFSK When the frequencies f 1 and f 2 are chosen in such a
way that they differ from each other by an amount equal to the
reciprocal of the bit duration T b
Slide 84
84
Slide 85
85 f c = 8 Hz, T b = 1s f = f c 1/(2T b ) BW = 3/T b
Slide 86
86 2/T b 3/T b for f 1,2 = f c 1/(2T b ) Bandwidth QPSK 1/T
b
Slide 87
87 M-ary Digital Modulation Schemes We send any one of M
possible signals during each signaling interval of duration T. The
requirement is to conserve bandwidth at the expense of both
increased power and increased system complexity. When the bandwidth
of the channel is less than the required value, we resort to an
M-ary modulation scheme for maximum bandwidth conservation
Slide 88
88 M-ary Phase-Shift Keying If we take blocks of m bits to
produce a symbol and use an M-ary PSK scheme with M=2 m and symbol
duration T=mT b The bandwidth required is proportional to 1/(mT b
). The use of M-ary PSK provides a reduction in transmission
bandwidth by a factor m=log 2 M over BPSK.
Slide 89
89
Slide 90
90 M-ary Quadrature Amplitude Modulation (QAM) The mathematical
description of the new modulated signal The level parameter for
in-phase component and quadrature component are independent of each
other for all i. M-ary QAM is a hybrid form of M-ary modulation.
M-ary amplitude-shift keying (M-ary ASK)
Slide 91
Signal-Space Diagram Figure 7.21 is the signal-space
representation of M-ary QAM for M=16 Unlike M-ary PSK, the
different signal points of M-ary QAM are characterized by different
energy levels Each signal point in the constellation corresponds to
a specific quadbit
Slide 92
92
Slide 93
93 Bit Error Rate Average bit error rate (BER) Let n denote the
number of bit errors observed in a sequence of bits of length N;
then the relative frequency definition of BER is BER goal: For data
transmission over wireless channels, a bit error rate of 10 -5 to
10 -6 For video transmission, a BER of 10 -7 to 10 -12 depending
upon the quality desired and the encoding method.
Slide 94
Signal to Noise Ratio (SNR) The ratio of the modulated energy
per information bit to the one-sided noise spectral density;
namely, The reference SNR is independent of transmission rate.
Since it is a ratio of energies, it has essentially been normalized
by the bit rate.
Slide 95
95 where
Slide 96
P e = 0.5P(0 decided| 1 is trans.) + 0.5P(1 decided|0 is
trans.) BASK
Slide 97
97
Slide 98
Real-world use (Tidbits) The wireless LAN standard, IEEE
802.11b-1999, uses a variety of different PSKs depending on the
data-rate required. - Basic-rate of 1 Mbit/s, DBPSK - Extended-rate
of 2 Mbit/s, DQPSK - 5.5 Mbit/s and the full-rate of 11 Mbit/s,
QPSK is used with complementary code keying. The higher-speed
wireless LAN standard, IEEE 802.11g- 2003 [1][3] has eight data
rates: 6, 9, 12, 18, 24, 36, 48 and 54 Mbit/s. [1][3] - The 6 and 9
Mbit/s modes, OFDM modulation where each sub-carrier is BPSK
modulated. - The 12 and 18 Mbit/s modes use OFDM with QPSK. - The
fastest four modes use OFDM with QAM
Slide 99
BPSK is appropriate for low-cost passive transmitters, and is
used in RFID standards. Bluetooth uses /4-DQPSK for the rate 2
Mbit/s and 8-DPSK at its higher rate (3 Mbit/s) IEEE 802.15.4 (the
wireless standard used by ZigBee) also relies on PSK. It has two
frequency bands: - 868915 MHz using BPSK and - 2.4 GHz using
OQPSK
Slide 100
References: Simon Haykin and Michael Moher, Introduction to
Analog and Digital Communications, 2 nd ed., John Wiley & Sons,
Inc., 2007. Charles L. Phillips, John M. Parr, Eve A. Riskin,
Signals, Systems, and Transforms, 4 th ed., Pearson/Prentice Hall,
2008.