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Matlab代写-ELEC6217
时间：2021-11-07
ELEC6217 Wireless Transceiver Design and Implementation
Modulation and Transmission
https://secure.ecs.soton.ac.uk/notes/elec6217/
Mohammed El-Hajjar
Bdg. 53, Room 4011
Tel.: 27032
email: meh@ecs.soton.ac.uk
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Introduction
• A communications system can be represented by the following block diagram:
Modulator
Demodulator
Ch
an
ne
l
Decoder
Channel
Encoder
Decoder
Source
Encoder
Source
Demapper
Mapper
Sink
Source Channel DAC
ADC
Digital Domain
ADC
DAC
ADC: Analogue-to-Digital Convertor. DAC: Digital-to-Analogue Convertor.
• The components of such a communications system involve signals and their
processing in each sub-block.
1
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Transmission
• A signal needs to be analogue and modulated to a Radio-Frequency (RF) carrier
with (angular) frequency ωc in order to be transmitted via antennas:
lation
modu− demodu−
lation
interference
~~
DAC ADC
s(t)[n]x (t)x
(X jω)
ωc
(jω)S (X jω)
s(t)~ ~(t)x ~[n]x
ω ωω
• note: (i) different signal representations (time/frequency domain) are helpful;
(ii) distortion and noise influences need to be accounted and quantified.
2
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Analogue Modulation
• We want to transmit an analogue signal s(t) over a radio link;
• due to λ = c/f , for an electromagnetic wave to radiate from a reasonably sized
antenna, we need to convert s(t) into a high frequency signal sHF(t); this is done
by modulation:
lation
modu−s (t)
(t)HFs (t)HFr
(t)rdemodu−
lation
radio
channel
• at the receiver, the electromagnetic wave can be received by an antenna; this high
frequency signal rHF(t) needs to be demodulated back to an analogue signal r(t);
3
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Quadrature Amplitude Modulation
• Inphase and quadrature components can be two independent real-valued signals:
(ω t)sin c
(ω t)cos c(ω t)cos c
(ω t)sin c
( )ts
( )trs
( )tis
LP
LP
( )tr
( )ti
r
r
( )tr
( )tr
( )tr
HF
HF
HFr,
~
HF
~
i,
• the circuit on the right shows the demodulation stage, where ‘LP’ denotes the ‘Low
Pass’ filter.
4
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Baseband Model I
• Consider the modulation / physical channel / demodulation components of the
channel:
[ ]ns ( )tr [ ]nr( )tr( )ts ( )ts
complex−valued real−valued
(HF jω)R
ωc
ω
ωc
(HFS jω)
ωc
(jω)HFC(jω)S
ω
(jω)R
ωω ω
DAC ADCHFHF
mod.
QAM physical
channel
QAM
demod.
• the physical channel can be dispersive due to multipath propagation and potentially
time-varying due to antenna mobility or path changes:
( )ts ( )tr
( )tc
HF HF
HFchannelphysical
5
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Baseband Model II
• In a first step to obtaining a baseband model, we replace the up-modulated signal
filtered by the channel by our baseband signal filtered by the down-modulated
channel:
c( )t[ ]ns [ ]nr( )ts ( )tr
ωc
(jω)C
ω
(jω)S
ω
(jω)R
ω
DAC ADC
• in case of an ideal physical channel
cHF(t) = δ(t) → c(t) = δ(t) and r(t) = s(t)
since the modulation is transparent;
• to achieve r[n] = s[n], we will have to place special conditions on DAC and ADC.
6
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Baseband Model III
• Neglecting quantisation, consider a DAC with “reconstruction filter” g(t) and an
ADC with “anti-alias filter” f(t):
[ ]ns ( )ts ( )tr [ ]nr
c t( )
h n[ ]
Ts Ts
g )t( f )t(
RX filterTX filter
DAC ADC
• thus, the overall analogue baseband channel model is h(t) = g(t) ∗ c(t) ∗ f(t);
• the baseband model h[n] in r[n] = h[n] ∗ s[n] is a sampled version of h(t);
• to achieve h[n] = δ[n], the obvious choice h(t) = δ(t) is not the only solution . . .
7
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Nyquist System I
• Consider a system h(t) with the following magnitude response |H(jω)|:
ωs/2 ωs/2
symmetry point
(jω)H
ω
1/2
1
0
−
• when sampling h(t), the discrete result h[n] has periodicity (discrete waveform ◦—•
periodic spectrum) with respect to the sampling rate enforced in its spectrum H(ejΩ):
(H e jΩ)
1
Ω
pi 2pi 3pi
0
pi−
(f = fs/2) (f = fs)
cos(ωt)→ cos(ωn/fs) = cos(Ωn)
t→ nTs = n/fs
Ω = ω/fs = 2pif/fs
• to achieve a flat H(ejΩ) •—◦ δ[n], a sufficient condition for h(t) is point symmetry
of the transition band of H(jω) w.r.t. w = ws/2.
8
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Nyquist System II
• A system H(jω) yielding a flat frequency response in the sampling process is a
Nyquist system;
• in the time domain, a Nyquist system is characterised by regular zero crossing:
h(t) =
{
1 t = 0
0 t = ±nTs, n = 1, 2, . . . (1)
• we have no control over the physical channel, but we can design the Tx and Rx
filter accordingly, such that h(t) = g(t) ∗ f(t) is a Nyquist system;
• example: a raised cosine filter is split into two root raised cosine filters with
response G(jω) = F (jω) =
√
H(jω), such that G(jω) · F (jω) = H(jω) is a
raised cosine filter fulfilling Nyquist;
9
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Nyquist Systems — Time Domain
−10 −8 −6 −4 −2 0 2 4 6 8 10
−0.2
0
0.2
0.4
0.6
0.8
1
normalised time t/Ts
filt
er
im
pu
lse
re
sp
on
se
sinc
square pulse
raised cosine
• All filters have regular zero-crossings (=Nyquist system), but different support.
10
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Nyquist Systems — Frequency Domain
0 0.5 1 1.5 2 2.5
−60
−50
−40
−30
−20
−10
0
normalised angular frequency ω/ω
s
filt
er
m
ag
ni
tu
de
re
sp
on
se
sinc
square pulse
raised cosine
• Some filters produce considerable excess bandwidth beyond ωs/2 (symmetry point).
11
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Nyquist Systems — Raised Cosine
−5 −4 −3 −2 −1 0 1 2 3 4 5
−0.2
0
0.2
0.4
0.6
0.8
1
Normalised time; t/T
s
N
or
m
al
is
ed
Im
pu
lse
R
es
po
ns
e;
h
(t)
/T s
α=0.0
α=0.5
α=1.0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Normalised Frequency; ω/ω
s
M
ag
ni
tu
de
R
es
po
ns
e;
|H
(ej
ω
)|
α=0.0
α=0.5
α=1.0
• Raise Cosine impulses having various roll-off factor (α) values.
• Regular zero-crossings at sampling period (Nyquist system).
• Note the excess bandwidth beyond ωs/2 (symmetry point).
12
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Nyquist Systems — Raised Cosine (After sampling at receiver)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
α=0.5, after time−domain sampling at ω
s
Normalised Frequency; ω/ω
s
M
ag
ni
tu
de
R
es
po
ns
e;
|H
(e
j ω
)|
• The resultant frequency response after sampling in the time-domain at a frequency
of ωs for a Raise Cosine filter having α = 0.5.
• Raised Cosine Filter magnitude response: H(jω) = G(jω) · F (jω)
13
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
After Transmit and Receive filters — Time Domain
−2 −1 0 1 2 3 4 5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Normalised time t/T
s
Fi
lte
r I
m
pu
lse
R
es
po
ns
e
• Pulse shaping using ‘raised cosine filter’ for bit sequence {1, 0, 1,1, 1} at the
inphase branch of a QPSK modulator.
14
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Transmit and Receive Filters
• As mentioned on slide 9, Tx and Rx filters together form a Nyquist system:
c( )t( )ts ( )tr
( )tv
( )ts ( )tr
g )t( f )t(
(jω)S(jω)S (jω)R
jω)G( jω)(F
(jω)R~
ω ωω
~
~
~
ω
ω
ω
• the Tx filter g(t) has a lowpass characteristic G(jω) , which ensures that the signal
s(t) is suitably bandlimited;
• the Rx filter f(t) has the same characteristic, ensuring that out-of-band noise will
be suppressed in the received signal r˜(t) prior to sampling.
15
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Discrete Baseband Channel Model
[ ]ns ( )tr [ ]nr( )tr( )ts ( )ts
complex−valued real−valued
(HF jω)R
ωc
ω
ωc
(HFS jω)
ωc
(jω)HFC(jω)S
ω
(jω)R
ωω ω
DAC ADCHFHF
mod.
QAM physical
channel
QAM
demod.
Actual system: transmit/receive filters, modulation/demodulation and the physical
(passband) channel. It involves waveform analysis, time/frequency synchronisation
and filter design.
s[n]
v[n]
h[n]
r[n]
Discrete Baseband Channel model: simplified model thanks to Nyquist theorem. It
involves only complex numbers: r[n] = h[n] ∗ s[n] + v[n].
16
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Dispersive Channel
• Ideally c(t) = A δ(t −∆Ts), such that the baseband channel h[n] = A δ[n −∆]
only scales and delays the transmitted discrete samples (−→ r[n] = A s[n−∆]);
• if c(t) is dispersive (and not a Nyquist system), then the overall baseband model
h[n] will be dispersive:
( )tr( )ts
( )tc
h[n]
n
− +1Nn ][s[n]s −1n ][s −2n ][s
[n]r
h[ ]1 h[ ]2 [ −1]h Nh[ ]0
c( )t
Ts Ts Ts
t
2 3 50 1 4
2 3 5
Tx/R
x filt
erin
g &
sam
plin
g
model
baseband
HFHF
HF
17
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Inter-Symbol Interference
• If h[n] is dispersive, each received sample r[n] is a weighted sum of transmitted
samples (or symbols) s[n], s[n− 1], s[n− 2], etc.
− +1Nn ][s−1n ][s −2n ][s
h[ ]1 h[ ]2 [ −1]h N
[n]s
[n]r
h[ ]0
• this “blurring effect” is called inter-symbol interference (ISI), i.e. neighbouring
symbols interfere with the transmitted symbol that we want to receive at time n;
• ISI makes symbol detection difficult/impossible; a remedy is the use of an equaliser:
s[n]
- h[n] -
r[n]
q[n] -
rq[n]
h[n] ∗ q[n] = δ[n−∆]
• note: if cHF(t) is time-varying, then so will be h[n] (and the equaliser q[n]).
18
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Inter-Symbol Interference (Time domain)
• Consider the response of an ideal pulse shaping filter with regular zero crossings
and the same system in combination with the channel impulse response c(t):
0 1 2 3 4 5 6 7 8
−0.2
0
0.2
0.4
0.6
0.8
1
time / symbol periods
tx
rx
fi
lte
r *
c
ha
nn
el
• The system: Tx-filter – channel – Rx filter, has lost the property of a Nyquist
system; peaks of the function no longer coincide with zero crossings of neighbouring
pulses.
19
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Eye Diagram — Perfect Channel
• We are looking at stacked 2 symbol period intervals of the ‘real-part’ (in-phase) of
the demodulated signal r(t) in a QPSK scheme (after receive filter):
0 0.5 1 1.5
x 10−4
−1.5
−1
−0.5
0
0.5
1
1.5
Eye Diagram of Real(Received QPSK Signal) : No Noise, IdealChannel
time t (sec)
Re
al(r
(t))
• Symbol period is Ts = 100µs. Ideal sampling will sample the crossing points
r(t) = ±0.707 −→ clock/timing recovery (τ ≈ 0.5Ts = 50µs).
20
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Eye Diagram — Noisy Channel
• With channel noise at 14dB SNR, the eye diagram looks different:
0 0.5 1 1.5
x 10−4
−1.5
−1
−0.5
0
0.5
1
1.5
Eye Diagram of Real(Received QPSK Signal) : SNR=14dB, IdealChannel
time t (sec)
Re
al(r
(t))
• As long as the sampling points can be clearly determined and the eye is “open”,
the sampled signal will correctly resemble the transmitted signal. At higher noise
levels, misclassifications can occur if the eye is “closed”.
21
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Eye Diagram — Dispersive Channel
• The channel is now non-ideal with an impulse response c(t) = δ(t)− 12 ·δ(t−Ts/4),
where Ts is the symbol period:
0 0.5 1 1.5
x 10−4
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Eye Diagram of Real(Received QPSK Signal) : No Noise, Dispersive Channel
time t (sec)
Re
al(r
(t))
• the eye diagram is distorted; more severely distorting channels will lead to
misclassification due to inter-symbol interference.
22
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
QAM for Digital Data
• To transmit digital data, we can use digital-to-analogue (DAC) and analogue-to-
digital conversion (ADC) combined with QAM:
( )trr
( )tir
r[ ]nr
( )tr( )ts
( )tis
( )trs
[si ]n
r[ ]ns
[ ]nri
physical
channel
ADC
ADC
HFHF
DAC
DAC
QAM
mod.
QAM
demod.
• ideally, in the receiver we would like to retrieve r[n] = rr[n] + jri[n] such that
r[n] = s[n] = sr[n] + jsi[n] ;
• in practice, we will incur a non-ideal transfer path (h[n] = discrete baseband model)
r[n] = h[n] ∗ s[n] (2)
23
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Transfering Digital Data
• Transmission of a bit stream sb[m]:
sb[m]
-
symbol
mapper
-
s[n]
h[n] -+?
v[n]
-
r[n] symbol
detection
-
rb[m]
h[n] models transmit and receive filters, modulation/demodulation and the physical channels.
v[n]: AWGN of atmospheric nature or stems from other users transmitting in the same band &
time slot or the thermal noise at the receiver.
• the symbol mapper maps the input bit stream sb[m] to a sequence of complex-
valued symbols s[n] according to a look-up table:
[n]s
[s m]
[n]s
[n]s
[n]s
1
’00’ ’10’
’01’ ’11’
Im{ }
Re{ }
Im{ }
Re{ }
n
n
b
m
1
0
0 1 2 3 4 5 6 7 8 9
24
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Constellation Diagrams
• The set of legitimate complex symbol values s[n] is called a constellation diagram;
• it can be understood as a look-up table for converting groups of bits into a complex
valued symbol;
• popular “digital modulation” (or rather ‘symbol mapper’) forms are:
[n]s
[n]s[n]s
[n]s
[n]s
[n]s
[n]s
[n]s
11
’0’ ’1’
’0100’
’0000’
’1111’
’1011’
1
’00’ ’10’
’01’ ’11’
Re{ }
Im{ }
’111’
’110’
’100’
’101’
’001’
’000’
’010’
’011’
Im{ }
Re{ } BPSK)(
8−PSK)(
Im{ }
Re{ }
Im{ }
Re{ }
QPSK( )
16−QAM
quadrature phase
shift keying
binary phase shift keying shift keying
8−level Phase
25
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Detection
• In the receiver, after Rx filtering and sampling, the correct symbol needs to be
detected;
[n]r
[n]r
1
’00’ ’10’
’01’ ’11’
Re{ }
Im{ }
– in case the received value deviates from a valid
constellation point, we have to estimate e.g. the
most likely transmitted symbol;
– in QPSK, we might assign all received values in
one quadrant to the constellation point within it,
i.e. any value in the shaded area will be mapped
to ’11’ (“decision thresholds”);
[m]r[n]
[n]
r
r
sgn( )
sgn( ) 1
’00’ ’10’
’01’ ’11’
n
n
}Re{
Im{ }
Imag
Real
b
m
1
0
0 1 2 3 4 5 6 7 8 9
0 1 2
31
5
26
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Noise
In communication systems, we tend to refer to unwanted waves that disturb the
transmission and processing of the signals as noise. We have no control over this
noise.
The noise analysis of communications systems is mostly based on an idealised form of
noise called white noise, whose power spectral density (PSD) is independent of the
operating frequency. We express the PSD of white noise as N02 .
f
N0/2
We model the noise in communications systems as Additive White Gaussian Noise.
27
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Additive noise in the frequency domain
• Recall that we assume the model of additive noise;
[n]s
[n]v
[n]r
(e jΩ)Rvv
Ω
R e jΩ)(rr
Ω
(e jΩ)Rss
Ω
• s[n] is the transmitted signal,
r[n] the received signal;
• the additive noise v[n] distorts
the received signal;
• measure signal quality: signal-
to-noise ratio (SNR);
• signal and noise power can be
determined from the PSDs.
28
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Signal-to-noise ratio (SNR)
The signal to noise ratio is an instantaneous power ratio:
SNR =
signal power at time n
noise power at time n
For zero-mean signals, SNR = σ2signal/σ
2
noise.
Since the range of values to be measured may span several orders of magnitude, we
use a logarithmic scale (dB) to measure
SNRdB = 10 · log10
σ2signal
σ2noise
= 20 · log10
σsignal
σnoise
[deciBel, dB]
Examples: σ2signal = 1, 000 · σ2noise −→ SNRdB = 30 dB;
σ2signal = 1, 000, 000 · σ2noise −→ SNRdB = 60 dB.
SNR versus Eb/N0?
29
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Influence of Channel Noise on QPSK
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
Re{r[n]}
Im
{r[
n]}
←− no channel
noise
some
channel
noise
−→
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
Re{r[n]}
Im
{r[
n]}
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
Re{r[n]}
Im
{r[
n]}
←−
stronger
channel
noisess
severe
channel
noise
−→
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
Re{r[n]}
Im
{r[
n]}
30
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Symbol Errors and Bit Errors
• Channel noise is likely to lift transmitted symbols across a decision threshold and
trigger some incorrect decisions — and finally bit errors — in the receiver;
• with constellations encoding several bits into one symbol, a symbol error translates
into potentially several bit errors;
• Gray coding with a change of only a single bit between adjacent points in the
constellation diagram ensures in most cases that one symbol error only leads to
one bit error;
• to quantify the noise influence, we want to calculate the bit error probability (ratio),
BER = (the number of erroneous bits)/(the total number of transmited bits).
31
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Bit Error Ratio (BER)
• We assume a non-dispersive channel h[n] = δ[n] and thus r[n] = s[n] + v[n] with
v[n] = N (0, σ2v):
sb[m]
-
symbol
mapper
-
s[n]
h[n] -+?
v[n]
-
r[n] symbol
detection
-
rb[m]
• the SNR at the symbol detector’s input is σ2s/σ2v;
• the BER can be calculated from the PDFs / CDFs of r[n] given s[n] (example for
BPSK with decision threshold r = 0):
r sr s =1)p ( |=−1)p ( |probability of
an erroneously
given a transmission
of a ’1’
detected ’−1’
an erroneously
probability of
detected ’1’
given a transmission
of a ’−1’d
−d
r
32
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Coursework
• These slides and the assignment are available from:
https://secure.ecs.soton.ac.uk/notes/elec6217/
• Mark contribution to the module: 30%
• submit your report electronically at C-BASS
https://handin.ecs.soton.ac.uk/handin/2122/ELEC6217/1/
before 4pm on Friday 29/10/2021.
• You only need to make an electronic submission; no need to print your coursework.
33
学霸联盟
Modulation and Transmission
https://secure.ecs.soton.ac.uk/notes/elec6217/
Mohammed El-Hajjar
Bdg. 53, Room 4011
Tel.: 27032
email: meh@ecs.soton.ac.uk
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Introduction
• A communications system can be represented by the following block diagram:
Modulator
Demodulator
Ch
an
ne
l
Decoder
Channel
Encoder
Decoder
Source
Encoder
Source
Demapper
Mapper
Sink
Source Channel DAC
ADC
Digital Domain
ADC
DAC
ADC: Analogue-to-Digital Convertor. DAC: Digital-to-Analogue Convertor.
• The components of such a communications system involve signals and their
processing in each sub-block.
1
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Transmission
• A signal needs to be analogue and modulated to a Radio-Frequency (RF) carrier
with (angular) frequency ωc in order to be transmitted via antennas:
lation
modu− demodu−
lation
interference
~~
DAC ADC
s(t)[n]x (t)x
(X jω)
ωc
(jω)S (X jω)
s(t)~ ~(t)x ~[n]x
ω ωω
• note: (i) different signal representations (time/frequency domain) are helpful;
(ii) distortion and noise influences need to be accounted and quantified.
2
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Analogue Modulation
• We want to transmit an analogue signal s(t) over a radio link;
• due to λ = c/f , for an electromagnetic wave to radiate from a reasonably sized
antenna, we need to convert s(t) into a high frequency signal sHF(t); this is done
by modulation:
lation
modu−s (t)
(t)HFs (t)HFr
(t)rdemodu−
lation
radio
channel
• at the receiver, the electromagnetic wave can be received by an antenna; this high
frequency signal rHF(t) needs to be demodulated back to an analogue signal r(t);
3
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Quadrature Amplitude Modulation
• Inphase and quadrature components can be two independent real-valued signals:
(ω t)sin c
(ω t)cos c(ω t)cos c
(ω t)sin c
( )ts
( )trs
( )tis
LP
LP
( )tr
( )ti
r
r
( )tr
( )tr
( )tr
HF
HF
HFr,
~
HF
~
i,
• the circuit on the right shows the demodulation stage, where ‘LP’ denotes the ‘Low
Pass’ filter.
4
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Baseband Model I
• Consider the modulation / physical channel / demodulation components of the
channel:
[ ]ns ( )tr [ ]nr( )tr( )ts ( )ts
complex−valued real−valued
(HF jω)R
ωc
ω
ωc
(HFS jω)
ωc
(jω)HFC(jω)S
ω
(jω)R
ωω ω
DAC ADCHFHF
mod.
QAM physical
channel
QAM
demod.
• the physical channel can be dispersive due to multipath propagation and potentially
time-varying due to antenna mobility or path changes:
( )ts ( )tr
( )tc
HF HF
HFchannelphysical
5
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Baseband Model II
• In a first step to obtaining a baseband model, we replace the up-modulated signal
filtered by the channel by our baseband signal filtered by the down-modulated
channel:
c( )t[ ]ns [ ]nr( )ts ( )tr
ωc
(jω)C
ω
(jω)S
ω
(jω)R
ω
DAC ADC
• in case of an ideal physical channel
cHF(t) = δ(t) → c(t) = δ(t) and r(t) = s(t)
since the modulation is transparent;
• to achieve r[n] = s[n], we will have to place special conditions on DAC and ADC.
6
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Baseband Model III
• Neglecting quantisation, consider a DAC with “reconstruction filter” g(t) and an
ADC with “anti-alias filter” f(t):
[ ]ns ( )ts ( )tr [ ]nr
c t( )
h n[ ]
Ts Ts
g )t( f )t(
RX filterTX filter
DAC ADC
• thus, the overall analogue baseband channel model is h(t) = g(t) ∗ c(t) ∗ f(t);
• the baseband model h[n] in r[n] = h[n] ∗ s[n] is a sampled version of h(t);
• to achieve h[n] = δ[n], the obvious choice h(t) = δ(t) is not the only solution . . .
7
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Nyquist System I
• Consider a system h(t) with the following magnitude response |H(jω)|:
ωs/2 ωs/2
symmetry point
(jω)H
ω
1/2
1
0
−
• when sampling h(t), the discrete result h[n] has periodicity (discrete waveform ◦—•
periodic spectrum) with respect to the sampling rate enforced in its spectrum H(ejΩ):
(H e jΩ)
1
Ω
pi 2pi 3pi
0
pi−
(f = fs/2) (f = fs)
cos(ωt)→ cos(ωn/fs) = cos(Ωn)
t→ nTs = n/fs
Ω = ω/fs = 2pif/fs
• to achieve a flat H(ejΩ) •—◦ δ[n], a sufficient condition for h(t) is point symmetry
of the transition band of H(jω) w.r.t. w = ws/2.
8
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Nyquist System II
• A system H(jω) yielding a flat frequency response in the sampling process is a
Nyquist system;
• in the time domain, a Nyquist system is characterised by regular zero crossing:
h(t) =
{
1 t = 0
0 t = ±nTs, n = 1, 2, . . . (1)
• we have no control over the physical channel, but we can design the Tx and Rx
filter accordingly, such that h(t) = g(t) ∗ f(t) is a Nyquist system;
• example: a raised cosine filter is split into two root raised cosine filters with
response G(jω) = F (jω) =
√
H(jω), such that G(jω) · F (jω) = H(jω) is a
raised cosine filter fulfilling Nyquist;
9
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Nyquist Systems — Time Domain
−10 −8 −6 −4 −2 0 2 4 6 8 10
−0.2
0
0.2
0.4
0.6
0.8
1
normalised time t/Ts
filt
er
im
pu
lse
re
sp
on
se
sinc
square pulse
raised cosine
• All filters have regular zero-crossings (=Nyquist system), but different support.
10
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Nyquist Systems — Frequency Domain
0 0.5 1 1.5 2 2.5
−60
−50
−40
−30
−20
−10
0
normalised angular frequency ω/ω
s
filt
er
m
ag
ni
tu
de
re
sp
on
se
sinc
square pulse
raised cosine
• Some filters produce considerable excess bandwidth beyond ωs/2 (symmetry point).
11
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Nyquist Systems — Raised Cosine
−5 −4 −3 −2 −1 0 1 2 3 4 5
−0.2
0
0.2
0.4
0.6
0.8
1
Normalised time; t/T
s
N
or
m
al
is
ed
Im
pu
lse
R
es
po
ns
e;
h
(t)
/T s
α=0.0
α=0.5
α=1.0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Normalised Frequency; ω/ω
s
M
ag
ni
tu
de
R
es
po
ns
e;
|H
(ej
ω
)|
α=0.0
α=0.5
α=1.0
• Raise Cosine impulses having various roll-off factor (α) values.
• Regular zero-crossings at sampling period (Nyquist system).
• Note the excess bandwidth beyond ωs/2 (symmetry point).
12
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Nyquist Systems — Raised Cosine (After sampling at receiver)
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
α=0.5, after time−domain sampling at ω
s
Normalised Frequency; ω/ω
s
M
ag
ni
tu
de
R
es
po
ns
e;
|H
(e
j ω
)|
• The resultant frequency response after sampling in the time-domain at a frequency
of ωs for a Raise Cosine filter having α = 0.5.
• Raised Cosine Filter magnitude response: H(jω) = G(jω) · F (jω)
13
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
After Transmit and Receive filters — Time Domain
−2 −1 0 1 2 3 4 5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Normalised time t/T
s
Fi
lte
r I
m
pu
lse
R
es
po
ns
e
• Pulse shaping using ‘raised cosine filter’ for bit sequence {1, 0, 1,1, 1} at the
inphase branch of a QPSK modulator.
14
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Transmit and Receive Filters
• As mentioned on slide 9, Tx and Rx filters together form a Nyquist system:
c( )t( )ts ( )tr
( )tv
( )ts ( )tr
g )t( f )t(
(jω)S(jω)S (jω)R
jω)G( jω)(F
(jω)R~
ω ωω
~
~
~
ω
ω
ω
• the Tx filter g(t) has a lowpass characteristic G(jω) , which ensures that the signal
s(t) is suitably bandlimited;
• the Rx filter f(t) has the same characteristic, ensuring that out-of-band noise will
be suppressed in the received signal r˜(t) prior to sampling.
15
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Discrete Baseband Channel Model
[ ]ns ( )tr [ ]nr( )tr( )ts ( )ts
complex−valued real−valued
(HF jω)R
ωc
ω
ωc
(HFS jω)
ωc
(jω)HFC(jω)S
ω
(jω)R
ωω ω
DAC ADCHFHF
mod.
QAM physical
channel
QAM
demod.
Actual system: transmit/receive filters, modulation/demodulation and the physical
(passband) channel. It involves waveform analysis, time/frequency synchronisation
and filter design.
s[n]
v[n]
h[n]
r[n]
Discrete Baseband Channel model: simplified model thanks to Nyquist theorem. It
involves only complex numbers: r[n] = h[n] ∗ s[n] + v[n].
16
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Dispersive Channel
• Ideally c(t) = A δ(t −∆Ts), such that the baseband channel h[n] = A δ[n −∆]
only scales and delays the transmitted discrete samples (−→ r[n] = A s[n−∆]);
• if c(t) is dispersive (and not a Nyquist system), then the overall baseband model
h[n] will be dispersive:
( )tr( )ts
( )tc
h[n]
n
− +1Nn ][s[n]s −1n ][s −2n ][s
[n]r
h[ ]1 h[ ]2 [ −1]h Nh[ ]0
c( )t
Ts Ts Ts
t
2 3 50 1 4
2 3 5
Tx/R
x filt
erin
g &
sam
plin
g
model
baseband
HFHF
HF
17
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Inter-Symbol Interference
• If h[n] is dispersive, each received sample r[n] is a weighted sum of transmitted
samples (or symbols) s[n], s[n− 1], s[n− 2], etc.
− +1Nn ][s−1n ][s −2n ][s
h[ ]1 h[ ]2 [ −1]h N
[n]s
[n]r
h[ ]0
• this “blurring effect” is called inter-symbol interference (ISI), i.e. neighbouring
symbols interfere with the transmitted symbol that we want to receive at time n;
• ISI makes symbol detection difficult/impossible; a remedy is the use of an equaliser:
s[n]
- h[n] -
r[n]
q[n] -
rq[n]
h[n] ∗ q[n] = δ[n−∆]
• note: if cHF(t) is time-varying, then so will be h[n] (and the equaliser q[n]).
18
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Inter-Symbol Interference (Time domain)
• Consider the response of an ideal pulse shaping filter with regular zero crossings
and the same system in combination with the channel impulse response c(t):
0 1 2 3 4 5 6 7 8
−0.2
0
0.2
0.4
0.6
0.8
1
time / symbol periods
tx
rx
fi
lte
r *
c
ha
nn
el
• The system: Tx-filter – channel – Rx filter, has lost the property of a Nyquist
system; peaks of the function no longer coincide with zero crossings of neighbouring
pulses.
19
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Eye Diagram — Perfect Channel
• We are looking at stacked 2 symbol period intervals of the ‘real-part’ (in-phase) of
the demodulated signal r(t) in a QPSK scheme (after receive filter):
0 0.5 1 1.5
x 10−4
−1.5
−1
−0.5
0
0.5
1
1.5
Eye Diagram of Real(Received QPSK Signal) : No Noise, IdealChannel
time t (sec)
Re
al(r
(t))
• Symbol period is Ts = 100µs. Ideal sampling will sample the crossing points
r(t) = ±0.707 −→ clock/timing recovery (τ ≈ 0.5Ts = 50µs).
20
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Eye Diagram — Noisy Channel
• With channel noise at 14dB SNR, the eye diagram looks different:
0 0.5 1 1.5
x 10−4
−1.5
−1
−0.5
0
0.5
1
1.5
Eye Diagram of Real(Received QPSK Signal) : SNR=14dB, IdealChannel
time t (sec)
Re
al(r
(t))
• As long as the sampling points can be clearly determined and the eye is “open”,
the sampled signal will correctly resemble the transmitted signal. At higher noise
levels, misclassifications can occur if the eye is “closed”.
21
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Eye Diagram — Dispersive Channel
• The channel is now non-ideal with an impulse response c(t) = δ(t)− 12 ·δ(t−Ts/4),
where Ts is the symbol period:
0 0.5 1 1.5
x 10−4
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Eye Diagram of Real(Received QPSK Signal) : No Noise, Dispersive Channel
time t (sec)
Re
al(r
(t))
• the eye diagram is distorted; more severely distorting channels will lead to
misclassification due to inter-symbol interference.
22
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
QAM for Digital Data
• To transmit digital data, we can use digital-to-analogue (DAC) and analogue-to-
digital conversion (ADC) combined with QAM:
( )trr
( )tir
r[ ]nr
( )tr( )ts
( )tis
( )trs
[si ]n
r[ ]ns
[ ]nri
physical
channel
ADC
ADC
HFHF
DAC
DAC
QAM
mod.
QAM
demod.
• ideally, in the receiver we would like to retrieve r[n] = rr[n] + jri[n] such that
r[n] = s[n] = sr[n] + jsi[n] ;
• in practice, we will incur a non-ideal transfer path (h[n] = discrete baseband model)
r[n] = h[n] ∗ s[n] (2)
23
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Transfering Digital Data
• Transmission of a bit stream sb[m]:
sb[m]
-
symbol
mapper
-
s[n]
h[n] -+?
v[n]
-
r[n] symbol
detection
-
rb[m]
h[n] models transmit and receive filters, modulation/demodulation and the physical channels.
v[n]: AWGN of atmospheric nature or stems from other users transmitting in the same band &
time slot or the thermal noise at the receiver.
• the symbol mapper maps the input bit stream sb[m] to a sequence of complex-
valued symbols s[n] according to a look-up table:
[n]s
[s m]
[n]s
[n]s
[n]s
1
’00’ ’10’
’01’ ’11’
Im{ }
Re{ }
Im{ }
Re{ }
n
n
b
m
1
0
0 1 2 3 4 5 6 7 8 9
24
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Constellation Diagrams
• The set of legitimate complex symbol values s[n] is called a constellation diagram;
• it can be understood as a look-up table for converting groups of bits into a complex
valued symbol;
• popular “digital modulation” (or rather ‘symbol mapper’) forms are:
[n]s
[n]s[n]s
[n]s
[n]s
[n]s
[n]s
[n]s
11
’0’ ’1’
’0100’
’0000’
’1111’
’1011’
1
’00’ ’10’
’01’ ’11’
Re{ }
Im{ }
’111’
’110’
’100’
’101’
’001’
’000’
’010’
’011’
Im{ }
Re{ } BPSK)(
8−PSK)(
Im{ }
Re{ }
Im{ }
Re{ }
QPSK( )
16−QAM
quadrature phase
shift keying
binary phase shift keying shift keying
8−level Phase
25
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Detection
• In the receiver, after Rx filtering and sampling, the correct symbol needs to be
detected;
[n]r
[n]r
1
’00’ ’10’
’01’ ’11’
Re{ }
Im{ }
– in case the received value deviates from a valid
constellation point, we have to estimate e.g. the
most likely transmitted symbol;
– in QPSK, we might assign all received values in
one quadrant to the constellation point within it,
i.e. any value in the shaded area will be mapped
to ’11’ (“decision thresholds”);
[m]r[n]
[n]
r
r
sgn( )
sgn( ) 1
’00’ ’10’
’01’ ’11’
n
n
}Re{
Im{ }
Imag
Real
b
m
1
0
0 1 2 3 4 5 6 7 8 9
0 1 2
31
5
26
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Noise
In communication systems, we tend to refer to unwanted waves that disturb the
transmission and processing of the signals as noise. We have no control over this
noise.
The noise analysis of communications systems is mostly based on an idealised form of
noise called white noise, whose power spectral density (PSD) is independent of the
operating frequency. We express the PSD of white noise as N02 .
f
N0/2
We model the noise in communications systems as Additive White Gaussian Noise.
27
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Additive noise in the frequency domain
• Recall that we assume the model of additive noise;
[n]s
[n]v
[n]r
(e jΩ)Rvv
Ω
R e jΩ)(rr
Ω
(e jΩ)Rss
Ω
• s[n] is the transmitted signal,
r[n] the received signal;
• the additive noise v[n] distorts
the received signal;
• measure signal quality: signal-
to-noise ratio (SNR);
• signal and noise power can be
determined from the PSDs.
28
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Signal-to-noise ratio (SNR)
The signal to noise ratio is an instantaneous power ratio:
SNR =
signal power at time n
noise power at time n
For zero-mean signals, SNR = σ2signal/σ
2
noise.
Since the range of values to be measured may span several orders of magnitude, we
use a logarithmic scale (dB) to measure
SNRdB = 10 · log10
σ2signal
σ2noise
= 20 · log10
σsignal
σnoise
[deciBel, dB]
Examples: σ2signal = 1, 000 · σ2noise −→ SNRdB = 30 dB;
σ2signal = 1, 000, 000 · σ2noise −→ SNRdB = 60 dB.
SNR versus Eb/N0?
29
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Influence of Channel Noise on QPSK
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
Re{r[n]}
Im
{r[
n]}
←− no channel
noise
some
channel
noise
−→
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
Re{r[n]}
Im
{r[
n]}
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
Re{r[n]}
Im
{r[
n]}
←−
stronger
channel
noisess
severe
channel
noise
−→
−1 0 1
−1.5
−1
−0.5
0
0.5
1
1.5
Re{r[n]}
Im
{r[
n]}
30
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Symbol Errors and Bit Errors
• Channel noise is likely to lift transmitted symbols across a decision threshold and
trigger some incorrect decisions — and finally bit errors — in the receiver;
• with constellations encoding several bits into one symbol, a symbol error translates
into potentially several bit errors;
• Gray coding with a change of only a single bit between adjacent points in the
constellation diagram ensures in most cases that one symbol error only leads to
one bit error;
• to quantify the noise influence, we want to calculate the bit error probability (ratio),
BER = (the number of erroneous bits)/(the total number of transmited bits).
31
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Bit Error Ratio (BER)
• We assume a non-dispersive channel h[n] = δ[n] and thus r[n] = s[n] + v[n] with
v[n] = N (0, σ2v):
sb[m]
-
symbol
mapper
-
s[n]
h[n] -+?
v[n]
-
r[n] symbol
detection
-
rb[m]
• the SNR at the symbol detector’s input is σ2s/σ2v;
• the BER can be calculated from the PDFs / CDFs of r[n] given s[n] (example for
BPSK with decision threshold r = 0):
r sr s =1)p ( |=−1)p ( |probability of
an erroneously
given a transmission
of a ’1’
detected ’−1’
an erroneously
probability of
detected ’1’
given a transmission
of a ’−1’d
−d
r
32
Wireless Transceiver Design and Implementation M. El-Hajjar, meh@ecs.soton.ac.uk
Coursework
• These slides and the assignment are available from:
https://secure.ecs.soton.ac.uk/notes/elec6217/
• Mark contribution to the module: 30%
• submit your report electronically at C-BASS
https://handin.ecs.soton.ac.uk/handin/2122/ELEC6217/1/
before 4pm on Friday 29/10/2021.
• You only need to make an electronic submission; no need to print your coursework.
33
学霸联盟
