Why modulation
What is modulation?
•Modulation is a process of imposing low-frequency information
onto a high-frequency carrier for transmission.
1. Baseband signal frequency- less .Length of antenna
required is large to transmit.
2.Channel characteristics – filtering effect- attenuation may
be more in LF band than RF.- depends on application
3. Multiplexing – Music, speech, voice… all are in
baseband, It becomes difficult to multiplex them with
image/video . Ex TV band, mobile band on same EM
channel- space, but different frequency band.
4Detection and estimation becomes less complex.
Signal representation-bandpass- to
baseband-
MODULATION
–PUT
BASEBAND
SIGNAL ON
CARRIER
But processing
signal in PB is
difficult and
costly.
Digital Modulation involve translating (recovering) the
baseband digital information to (from) a bandpass analog
signal at a carrier frequency that is very high compared
to the baseband frequency.
Digital modulation involves choosing a particular analog
signal waveform si(t) from a finite set S of possible
signal waveforms based on the information bits applied
to the modulator and imposing on basis.
For binary modulation only 2 signals, representing 0
and 1.
For M-ary modulations, more than 2 signals and each
represents more than a single bit of information. With a
signal set of size M, it is possible to transmit up to log2M
bits per signal.
Transmission of a Digital Message
Basically, it is the same as the analog modulation methods.
ASK: Amplitude shift keying
V(t) = A(t) cos(2πfc t + Φ)
FSK: Frequency shift keying
V(t) = A(t) cos(2πf(t) t + Φ)
PSK: Phase shift keying
V(t) = A(t) cos(2πf(t) t + Φ(t))
Digital modulation: amplitude, frequency, and/or phase are used to
represent a digital state.
‘The major concern is the optimum design of the receiver so as to
minimize the probability of symbol error in the presence of channel noise.
Bandwidth efficient and power
efficient modulation schemes
Modulation schemes categorized as either
a) bandwidth efficient or b) power efficient.
Bandwidth efficiency means that a modulation scheme is able to
accommodate more information (measured in bits/sec) per unit
(Hz) transmission bandwidth (e.g. 8-PSK) . Bandwidth efficient
modulation schemes are preferred more in digital terrestrial
microwave radios, satellite communications and cellular
telephony.
Power efficiency means the ability of a modulation scheme to
reliably send information at low energy per information bit.
Some cellular telephony systems and some frequency-hopping
spread spectrum communication systems (spread-spectrum
systems are wideband type) operate on power-efficient
modulation schemes.
Demod and detect
Signal demodulator: Vectorization (no information loss)
x(t) ⇒ [x1,x2,⋯,xN]
Detector : Minimize the probability of error in the above
functional block [x1,x2,⋯,xN] ⇒ estimator –si,detect mi
Two orthogonal but dependent signals-summed together
(i.e., simultaneously transmitted), can be completely
separated by communication system (if we know the basis).
Two independent signals, summed together, cannot be
completely separated with probability one (by “inner
product” technology), if they are not orthogonal to each
other. Therefore, in practice, orthogonality is more essential
than independence..
Geometric representation
Optimum RX
Design of optimum Rx is based on
1. Minimisation of Probability of bit error
2. optimisation of SNR
Optimum RX
X
The principle of Maximum Likelihood (ML) detection provides a
general solution to an optimum receiver – Correlator RX
ML detection operates on received x(t) transfprmed as vector X
to produce estimate of transmitted symbol mi; i=1 to M to
reduce Pe
The N vector elements of received observation vector X is first
multiplied by corresponding elements of M signal vectors s1,
s2….sM and resulting products are successively summed in
accumulator to for the to form set of linear products {(x,sK)}.
The inner products are adjusted by correcting for energies which
may be unequal.
The largest in the resulting set is selected and corresponding
decision on the message is made.
P[x|so sent]
P[x|s1| sent]
Matched Filter
In communication systems problem that arises is in detecting a
pulse transmitted over a channel that is corrupted by noise
Matched filter Rx is based on maximisation of SNR at the
receiver. It passes all the signal frequency components while
suppressing any frequency components of noise. To optimize the
design of the filter so as to minimize the effects of noise at the
filter output and improve the detection of the pulse
g(t) x(t) y(t) y(T)
h(t)
Pulse t=T
signal Matched
w(t) filter
Additive white Gaussian noise
(AWGN) with zero mean and y (t ) g (t ) * h(t ) w(t ) * h(t )
variance N0 /2 g 0 (t ) n(t )
Matched Filter Derivation
Design:
Maximize signal power i.e. power of g0 (t ) g (t ) * h(t ) at t = T
Minimize noise i.e. power of n(t ) w(t ) * h(t )
Combine design criteria
max SNR,
| g 0 (T ) |2 instantaneous power
SNR 2
E{n (t )} average noise power
g(t) x(t) y(t) y(T)
h(t)
Pulse t=T
signal Matched
w(t) filter
Matched Filter Derivation
Noise power
g(t) x(t) y(t) y(T) spectrum SW(f)
h(t)
N0
Pulse t=T
2
signal w(t) Matched filter f
Noise
N0
E{ n 2 (t ) } S N ( f ) df | H ( f ) |
2
df
Signal
2
g0 (t ) g (t ) * h(t ) G0 ( f ) H ( f )G( f )
H ( f ) G( f ) e
j 2 f t
g 0 (t ) df
H ( f ) G( f ) e
j 2 f T
| g 0 (T ) |2 | df |2
Matched Filter Derivation
We have to find h(t) that maximizes pulse peak SNR
H ( f ) G( f ) e
j 2 f T
| df |2
SNR
N0
| H ( f ) |
2
df
2
Applying Schwartz’s inequality for complex functions
For functions:
2
( x) ( x) ( x)
2 2
1
*
2 ( x) dx 1 dx 2 dx
- - -
If following condition
satisfies 1 ( x) k2* ( x) (4.10)
Matched Filter Derivation
Let 1 ( f ) H ( f ) and 2 ( f ) G * ( f ) e j 2 f T
| H ( f ) G( f ) e j 2 f T
df | | H ( f ) | df
2 2
| G( f ) |
2
df
-
| H ( f ) G ( f ) e j 2 f T df |2
2
SNR -
| G( f ) |
2
df
N0 N0
| H( f )|
2
df
2
2
SNRmax | G( f ) |
2
df , which occurs when
N0 Except for a scaling coefficient
j 2 f T k exp(-2πfT), the frequency response of
H opt ( f ) k G ( f ) e
*
the optimum filter is the same as the
complex conjugate of the FT of the input
signal.
Matched Filter
Given transmitter pulse shape g(t) of duration T, matched filter is
given by hopt(t) = k g*(T-t) for all k
Duration and shape of impulse response of the optimal filter is
determined by pulse shape g(t)
hopt(t) is scaled, time-reversed, and shifted version of g(t) hence defined
MF
Optimal filter maximizes peak pulse SNR
2 2 2 Eb
| G( f ) | df N0 | g (t ) | dt SNR
2 2
SNRmax
N0
N0
Does not depend on pulse shape g(t)
Proportional to signal energy (energy per bit) Eb
Inversely proportional to power spectral density of noise
h1(t)
g1
h2(t)
g2
hN(t)
gN
Properties of Matched Filters
Property 1:
A filter matched to a pulse signal g(t) of duration T is
characterized by an impulse response which is time-
reversed and delayed version of the input g(t):
Time domain: hopt(t) = k . g(T-t)
Frequency Domain: Hopt(f) = kG*(f)exp(-j2πfT)
Property 2:
The output of matched filter is proportional to a
shifted version of the autocorrelation function of the
input signal to which the filter is matched
g0 (t ) Rg (t - T)
Property 3:
The peak signal-to-noise ratio of the matched
filter depends only on the ratio of the signal
energy to the power spectral density of the white
noise at the filter input.
2 Eb
SNRmax
N0
Correlator versus Matched Filter
The mathematical operation of Correlator Rx
based on is correlation, where a received signal is
correlated with its replica of one sent
Whereas the operation of Matched filter is
Convolution, where signal is convolved with filter
impulse response
But the o/p of both is same at t=T
Implementation of matched filter
receiver
Bank of M matched filters
x1 (T )
s1 (T t )
*
x1
x Matched filter output:
x(t ) Observation
vector
sM (T t )
* x N
xN (T )
xi x(t ) si (T t ) i 1,..., N
Implementation of correlator
receiver
Bank of M correlators
s 1 (t )
T x1 (T )
0
x1 Correlators output:
x(t ) x Observation
s M (t ) vector
T x N
0 xN (T )
T
xi x(t )si (t )dt i 1,..., N
0
Classification
IQ Representation in Modulated
Carrier
V (t ) A(t ) cos[2f ct (t )] A(t ) cos (t ) cos 2f ct A(t ) sin (t ) sin 2f ct
I (t) Complex BB signal Q (t)
A(t ) I (t ) 2 Q(t ) 2 (t ) tan 1[Q(t ) / I (t )]
I component of V(t) I component of BB
Q
V(t)
Q (t) A (t)
(t )
Q component of BB I (t)
I
Q component of V(t)
Amplitude Shift Keying (ASK)
Baseband
Data
1 0 0 1 0
ASK
modulated
signal
Acos(t) Acos(t)
Pulse shaping can be employed to remove spectral spreading
as it is heavily affected by noise, prone to fading, and interference
Susceptible to sudden gain changes
On voice-grade lines, used up to 1200 bps
Used to transmit digital data over optical fiber
Frequency Shift Keying (FSK)
Baseband
Data
1 0 0 1
BFSK
modulated
signal
f1 f0 f0 f1
where f0 =Acos(c-)t and f1 =Acos(c+)t
FSK can be expanded to a M-ary scheme, employing multiple frequencies as different
states
1. Less susceptible to error than ASK
2. On voice-grade lines, used up to 1200bps
3. Used for high-frequency (3 to 30 MHz) radio transmission
4. Can be used at higher frequencies on 59
LANs that use coaxial cable
5. The ITU-T V.21 modem standard uses FSK
Frequency Shift Keying (FSK)
Baseband
Data
1 0 0 1
BFSK
modulated
signal
f1 f0 f0 f1
where f0 =Acos(c-)t and f1 =Acos(c+)t
FSK can be expanded to a M-ary scheme, employing multiple frequencies as different
states
1. Less susceptible to error than ASK
2. On voice-grade lines, used up to 1200bps
3. Used for high-frequency (3 to 30 MHz) radio transmission
4. Can be used at higher frequencies on 59
LANs that use coaxial cable
5. The ITU-T V.21 modem standard uses FSK
Draw the BASK,BFSK and BPSK waveform for
010011
Binary Code 0 1 0 0 1 1
t
(NRZ)
B-ASK t
B-FSK t
B-PSK t
Also termed as ON OFF Keying
Tb
BASK TR - RX operation
What will be the threshold level?
BPSK
Polar
form
IS X(t)
WIDEBAND OR
NARROWBAND?
It is NB-IF
BPSK constellation
At x1=0; z=√Eb/No
At x1=∞; z=∞
And dx1/√N0= dz
TR
FSK
QPSK
Efficient utilization of channel bandwidth is achieved by
bandwidth-conserving modulation scheme known as
quadriphase-shift keying (QPSK), using coherent detection.
phase of the carrier takes on one of four equally spaced values,
such as π/4, 3π/4, 5π/4, and 7π/4. For this set of values, we may
define the transmitted signal as
where E is the transmitted signal energy per symbol and T is the
symbol duration.
Signal space constellation
QPSK Constellation Diagram
Q Q
I I
Carrier phases Carrier phases
{0, /2, , 3/2} {/4, 3/4, 5/4, 7/4}
Quadrature Phase Shift Keying has twice the
bandwidth efficiency of BPSK since 2 bits are
transmitted in a single modulation symbol
Transmitter
Binary data sequence is first transformed into polar form by a
NRZ level encoder.
symbols 1& 0 are represented by +√ Eb and -√ Eb respectively.
Binary wave is next divided by means of a demultiplexer [Serial
to parallel conversion] into two separate binary waves consisting
of the odd and even numbered input bits. These two binary waves
are denoted by a1(t) and a2(t).
The two binary waves a1(t) and a2(t) are used to modulate a pair
of quadrature carriers or orthonormal basis functions Φ1(t) and
Φ2(t) .
The result is a pair of binary PSK signals, are added to produce
the desired QPSK signal.
vv
Draw QPSK waveform showing add and even
waveform and resultsnt QPSK
Review Question
Draw the ASK waveform for message 0101001 . Find the
occupied bandwidth for an ASK signal with raised cosine
pulse shaping, α = 0.5, as shown below:
Amplitude/volt
0 1 0 1 0 0 1
Time/s
Ans 1 second
Bit Rate =
Tx BW of ASK BT (1 ) Rb / 2 (1 0.5)7 / 2 5.25 Hz
PSK
FSK
Error probability of FSK
As M increases SNR
requirement reduces
Differential PSK
DPSK is viewed as the “noncoherent” version of binary PSK.
The distinguishing feature of DPSK is that it eliminates the need
for synchronizing the receiver to the transmitter by combining
Two basic operations at the transmitter:
• differential encoding of the input binary sequence and
• PSK of the encoded sequence
Differential encoding starts with an arbitrary first bit, serving as
the reference bit; to this end, symbol 1 is used as the reference bit.
Generation of the differentially encoded sequence
then proceeds in accordance with a two-part
encoding rule as follows:
1. If the new bit at the transmitter input is 1, leave
the differentially encoded symbol unchanged with
respect to the current bit.
2. If, on the other hand, the input bit is 0, change
the differentially encoded symbol with respect to
the current bit.
Generation of the differentially encoded sequence
then proceeds in accordance with a two-part
encoding rule as follows:
1. If the new bit at the transmitter input is 1, leave
the differentially encoded symbol unchanged with
respect to the current bit.
2. If, on the other hand, the input bit is 0, change
the differentially encoded symbol with respect to
the current bit.
The differentially encoded sequence, denoted by
{dk}, is used to shift the sinusoidal carrier phase
by zero and 180o, representing symbols 1 and 0,
respectively. Thus, in terms of phase-shifts, the
resulting DPSK signal follows the two-part rule:
1. To send symbol 1, the phase of the DPSK
signal remains unchanged.
2. To send symbol 0, the phase of the DPSK
signal is shifted by 180.
The differentially encoded sequence, denoted by
{dk}, is used to shift the sinusoidal carrier phase
by zero and 180o, representing symbols 1 and 0,
respectively. Thus, in terms of phase-shifts, the
resulting DPSK signal follows the two-part rule:
1. To send symbol 1, the phase of the DPSK
signal remains unchanged.
2. To send symbol 0, the phase of the DPSK
signal is shifted by 180.
ILLUSTRATION
Consider the input binary sequence, denoted {bk}, to be
10010011, which is used to derive the generation of a DPSK
signal.
The differentially encoded process starts with the reference bit 1.
Let {dk}denote the differentially encoded sequence and
{dk-1}denote its delayed version by one bit. The complement of
the modulo-2 sum of and defines the desired , as illustrated in the
Table. In the last line of this table, binary symbols 1 and 0 are
represented by phase-shifts of 1 and π radians.
Differential encoding: bk dk-1
At receiver
RX phase 0 0 π 0 0 π 0 0 0
Delayed phase
0 0 π 0 0 π 0 0
Estimated
1 0 0 1 0 0 1 1
Sequence {bk}
Generation –
Transmitter consists of two functional blocks:
• Logic network and one-bit delay (storage) element, which are
interconnected so as to convert the raw input binary sequence {bk} into the
differentially encoded sequence {dk}.
• Binary PSK modulator, the output of which is the desired DPSK signal.
RX
2.Binary data is transmitted over a microwave link at a rate
of 10Mbps and PSD of noise is 10-10
watts/hz. Find average carrier power required to maintain
average Pe <= 10-4 for coherent BFSK. What is required BW
Soln: Pe= 10-4 ,N0/2= 10-10
erfc == 2x 10-4
Therefore =2.6 or Eb = 6.7x2xN0
Eb=33.8x 10-10
Pav=Eb/Tb=336mW
So to achieve Pe >= 10-4 , it is required to maintain
Pav=336mW
B=1/Tb=1MHz
2) An FSK system transmits transmits binary data at the rate of
2.5 Mbps. During transmission white noise of PSD 10-4 watts/Hz
gets added to the signal. In the absence of noise amplitude of
received sinusoid for digit 1 or 0 is 1microvolt. Determine average
probability of symbol eror, assuming coherent detection.
Soln: Eb= a-2Tb/2 Tb=0.4microsec
Pe=9.3x 10-4
v
1. For any value of Eb/N0, the schemes using coherent
detection produce a smaller BER than those using
noncoherent detection.
2. binary PSK with coherent detection and DPSK with
noncoherent detection, require an Eb/N0 that is 3 dB less
than BFSK to realize the same BER.
3. At high values of Eb/N0, DPSK and binary FSK using
noncoherent detection perform almost as well, to within
about 1 dB of their respective counterparts using coherent
detection for the same BER.
OFDM
OFDM, which stands for Orthogonal Frequency Division
Multiplexing, is a modulation technique for transmitting
large amounts of digital data over a radio wave.
capable of achieving the quality multi-gigabit data rates
Orthogonal frequency-division multiplexing (OFDM) is a
method of digital modulation in which a signal is split into
several narrowband channels at different frequencies.
The technology was first conceived in the 1960s and 1970s
during research into minimizing interference among
channels near each other in frequency.
OFDM
OFDM, which stands for Orthogonal Frequency Division
Multiplexing, is a modulation technique for transmitting
large amounts of digital data over a radio wave.
capable of achieving the quality multi-gigabit data rates
Orthogonal frequency-division multiplexing (OFDM) is a
method of digital modulation in which a signal is split into
several narrowband channels at different frequencies.
The technology was first conceived in the 1960s and 1970s
during research into minimizing interference among
channels near each other in frequency.
OFDM
•To overcome inter-symbol interference ,
•Extended FDM-with Orthogonal subcarriers
•OFDM is combination of Modulation and Multiplexing
•Spectrally Efficient ,High data rate scheme
Frequency Division Multiplexing OFDM
Modulation and Multiplexing: Mapping of information on changes in the carrier
phase, frequency or amplitude or combination. Method of sharing bandwidth
with other independent data channels.
A Basic baseband OFDM system
Transmitter
Serial
Constella Serial
Bits tion to to
Add
mapping
IFFT
parall parallel Cyclic prefix
el
Parallel
Constellati to
Bits FFT Remove CP
on
Serial
Receiver
