Transcript OFDM Modulation - Home pages of ESAT

DSP-CIS
Chapter-3: Acoustic Modem Project
Marc Moonen
Dept. E.E./ESAT, KU Leuven
[email protected]
www.esat.kuleuven.be/scd/
Introduction
•
Will consider digital communications over acoustic channel:
Discrete-time
transmit signal
Discrete-time
receiver signal
(sampling rate Fs, e.g. 10kHz)
(sampling rate Fs, e.g. 10kHz)
Tx
D-to-A
+filtering
+amplif.
Digital Picture (IN)
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
A-to-D
Rx
+filtering
+…
Digital Picture (OUT)
p. 2
Introduction
•
Will consider digital communications over acoustic channel:
Discrete-time
transmit signal
Discrete-time
receiver signal
(sampling rate Fs, e.g. 10kHz)
(sampling rate Fs, e.g. 10kHz)
Tx
D-to-A
+filtering
+amplif.
A-to-D
Rx
+filtering
+…
This will be the easy part…
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 3
Introduction
•
Will consider digital communications over acoustic channel:
…straightforwardly realized
(in Matlab/Simulink with `Real-Time Workshop’, see below)
Discrete-time
transmit signal
Discrete-time
receiver signal
(sampling rate Fs, e.g. 10kHz)
(sampling rate Fs, e.g. 10kHz)
Tx
D-to-A
+filtering
+amplif.
A-to-D
Rx
+filtering
+…
means we do not have to deal with
hardware issues, components, etc.
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 4
Introduction
•
Will consider digital communications over acoustic channel:
…and will be modeled by a linear
Discrete-time discrete-time transfer function Discrete-time
transmit signal
(see below)
(sampling rate Fs, e.g. 10kHz)
(sampling rate Fs, e.g. 10kHz)
Tx
D-to-A
+filtering
+amplif.
receiver signal
H(z)
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
A-to-D
Rx
+filtering
+…
p. 5
Introduction
•
Will consider digital communications over acoustic channel:
Discrete-time
transmit signal
Discrete-time
receiver signal
(sampling rate Fs, e.g. 10kHz)
(sampling rate Fs, e.g. 10kHz)
Tx
D-to-A
+filtering
+amplif.
A-to-D
Rx
+filtering
+…
This is the interesting part…
(where we will spend most of the time)
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 6
•
Introduction
Will use OFDM as a modulation format
Orthogonal frequency-division multiplexing
From Wikipedia, the free encyclopedia
Orthogonal frequency-division multiplexing (OFDM), essentially identical to (…)
discrete multi-tone modulation (DMT), is a frequency-division multiplexing (FDM)
scheme used as a digital multi-carrier modulation method. A large number of closelyspaced orthogonal sub-carriers are used to carry data. The data is divided into several
parallel data streams or channels, one for each sub-carrier. Each sub-carrier is
modulated with a conventional modulation scheme (such as quadrature amplitude
modulation or phase-shift keying) at a low symbol rate, maintaining total data rates
similar to conventional single-carrier modulation schemes in the same bandwidth.
OFDM has developed into a popular scheme for wideband digital communication,
whether wireless or over copper wires, used in applications such as digital television
and audio broadcasting, wireless networking and broadband internet access.
-
OFDM/DMT is used in ADSL/VDSL, WiFi, DAB, DVB …
OFDM heavily relies on DSP functionalities (FFT/IFFT,
…)
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 7
Channel Modeling & Evaluation
Transmission channel consist of
–
–
–
–
–
Tx `front end’: filtering/amplification/Digital-to-Analog conv.
Loudspeaker (ps: cheap loudspeakers mostly have a non-linear characteristic )
Acoustic channel
Microphone
Rx `front end’: filtering/Analog-to-Digital conv.
Discrete-time
transmit signal
Discrete-time
receiver signal
(sampling rate Fs, e.g. 10kHz)
(sampling rate Fs, e.g. 10kHz)
Tx
D-to-A
+filtering
+amplif.
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
A-to-D
Rx
+filtering
+…
p. 8
Channel Modeling & Evaluation
Acoustic channel (`room acoustics’):
Acoustic path between loudspeaker and microphone is represented by
the acoustic impulse response (which can be recorded/measured)
–first there is a dead time
–then come the direct path impulse
and some early reflections, which
depend on the geometry of the room
–finally there is an exponentially decaying tail called reverberation,
corresponding to multiple reflections on walls, objects,...
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 9
Channel Modeling & Evaluation
-1
-2
H(z) = h0 + h1.z + h2.z +... + hL .z
-L
– Pragmatic & good-enough approximation
– Model order L depends on sampling rate (e.g. L=100…1000…)
Tx
D-to-A
+filtering
+amplif.
H(z)
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
A-to-D
+filtering
+…
Rx
PS: will use shorthand notation here, i.e.
hk, xk, yk , instead of h[k], x[k], y[k]
Complete transmission channel will be modeled by a
discrete-time (FIR `finite impulse response’) transfer function
p. 10
Channel Modeling & Evaluation
When a discrete-time (Tx) signal xk is sent over a channel…
H(z) = h0 + h1.z-1 + h2.z-2 +... + hL .z-L
..then channel output signal (=Rx input signal) yk is
é
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ë
yk
yk+1
yk+2
yk+3
yk+4
yk+K
ù é
ú ê
ú ê
ú ê
ú ê
ú=ê
ú ê
ú ê
ú ê
ú ê
ú
û ê
ë
xk
xk-1
xk-2
xk-L
xk+1
xk
xk-1
xk+1-L
xk+2
xk+1
xk
xk+2-L
xk+3
xk+2
xk+1
xk+3-L
xk+4
xk+3
xk+2
xk+4-L
xk+K
xk+K -1
xk+K -2
xk+K -L
ù
ú
úé
úê
úê
ú. ê
úê
úê
úê
ë
úê
ú
û
h0 ù
ú
h1 ú
ú
h2 ú
ú
ú
hL ú
û
=`convolution’
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 11
Channel Modeling & Evaluation
Can now run parameter estimation experiment:
1. Transmit `well-chosen’ signal xk
2. Record corresponding signal yk
yk
xk
Tx
D-to-A
H(z)
+filtering
+amplif.
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
A-to-D
Rx
+filtering
+…
p. 12
Channel Modeling & Evaluation
3. Least squares estimation
yk
yk+1
yk+2
yk+3
yk+4
yk+K
ù é
ú ê
ú ê
ú ê
ú ê
ú-ê
ú ê
ú ê
ú ê
ú ê
úû êë
xk
xk-1
xk-2
xk-L
xk+1
xk
xk-1
xk+1-L
xk+2
xk+1
xk
xk+2-L
xk+3
xk+2
xk+1
xk+2-L
xk+4
xk+3
xk+2
xk+4-L
xk+K
xk+K -1
xk+K -2
xk+K -L
ù
ú
úé
úê
úê
ú. ê
úê
úê
úê
ú êë
úû
h0 ù
ú
h1 ú
ú
h2 ú
ú
ú
hL ú
û
2
(i.e. one line of Matlab code )
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 13
Carl Friedrich Gauss (1777 – 1855)
min h0 ,h1,h2 ,...,hL
é
ê
ê
ê
ê
ê
ê
ê
ê
ê
êë
2
Channel Modeling & Evaluation
Estimated transmission channel can then be analysed…
• Frequency response
• Information theoretic capacity
f max
C ( bits/sec ) 
 log
f min
2
(1 
S( f )
) df
N(f )
ps: noise spectrum?
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013Claude
Shannon 1916-2001
p. 14
OFDM modulation
DMT – Discrete Multitone Modulation
OFDM – Orthogonal Frequency Division Multiplexing
Basic idea is to (QAM-)modulate (many) different carriers with low-rate bit
streams. The modulated carriers are summed and then transmitted.
A high-rate bit stream is thus carried by dividing it into hundreds
of low-rate streams.
Modulation/demodulation is performed by FFT/IFFT (see below)
Now 14 pages of (simple) maths/theory…
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 15
OFDM Modulation
1/14
Consider the modulation of
a complex exponential carrier (with period N)
ck = (e j 2 p /N )k for k = 0,1,...
carrier
ck
by a `symbol sequence’ (see p.21)
Xk for k = 0,1,...
defined as
xk = ck .X k
Xk
xk
x
ê k ú symbol
for k = 0,1,... and k = ê ú sequence
ëNû
(i.e. “1 symbol per N samples of the carrier”)
• PS: remember that modulation of sines and cosines is similar/related
to modulation of complex exponentials (see also p.20, 2nd ‘PS’)
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 16
OFDM Modulation
xk = ck .X k
êkú
for k = 0,1,... and k = ê ú
ëNû
carrier
ck
This corresponds to…
é
xk
ê
ê xk+1
ê
ê xk+2
ê xk+3
ê
ê xk+4
ê
ê
xk+N -1
ê
ë
2/14
ù é (e j 2 p / N )0
ú ê
j 2p /N 1
)
ú ê (e
ê
ú
j 2p /N 2
(e
)
ê
ú
ú = ê (e j 2 p / N )3
ú ê
j 2p /N 4
)
ú ê (e
ú ê
ú ê
ê (e j 2 p / N ) N -1
ú
û ê
ë
ù
ú
ú
ú
ú
ú.X
k
ú
ú
ú
ú
ú
ú
û
Xk
xk
x
symbol
sequence
time-domain
signal segment
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 17
OFDM Modulation
3/14
Now consider the modulation of
N such complex exponential carriers
x
…
ck(n) = (e j 2 p n/ N )k for k = 0,1,... and n = 0,1,.., N -1
by `symbol sequences’
Xk(n) for k = 0,1,... and n = 0,1,.., N -1
defined as
(n)
k
(n)
k
X
(n )
k
x (kn )
x
…
x = c .X
(n)
k
)
c (n
k
êkú
for k = 0,1,... and k = ê ú
ëNû
+
N -1
xk = å x (kn )
n=0
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
x
p. 18
OFDM Modulation
4/14
This corresponds to…
é
ê
ê
é x
ù ê
k
ê
ú ê
ê xk+1 ú ê
ê
ú ê
x
ê k+2 ú ê
ê xk+3 ú = ê
ê
ú ê
x
ê k+4 ú ê
ê
ú ê
ê
ú ê
x
êë k+N -1 úû ê
ê
time-domain
ê
signal segment
ê
ë
(e
(e
(e
(e
(e
(e
j.
2p
N 0
j.
2p
N 0
j.
2p
N 0
j.
2p
N 0
j.
2p
N 0
j.
2p
N 0
)
)
)
)
)
)
(e
(e
(e
(e
(e
(e
j.
j.
2p
N 0
)
j.
2p
N 1
)
j.
2p
N 2
j.
2p
N 3
j.
2p
N
)
)
2p
N
)
)
4
N-1
(e
(e
(e
(e
(e
(e
j.
j.
2p
N 0
j.
2p
N 2
j.
2p
N 4
j.
2p
N 6
j.
2p
N 8
)
)
)
)
)
2p
N 2( N-1)
)
(e
(e
(e
(e
(e
(e
j.
j.
2p
N 0
j.
2p
N 3
j.
2p
N 6
j.
2p
N 9
j.
)
)
)
)
2p
N 12
(e
(e
(e
(e
)
(e
2p
N 3( N -1)
j.
)
(e
j.
2p
N 0
j.
2p
N
j.
2p
N 8
)
)
4
)
j.
2p
N 12
j.
2p
N 16
)
2p
N
)
)
4( N -1)
…
…
…
…
…
…
j.
(e
(e
(e
(e
(e
j.
2p
N 0
)
2p
N
)
N -1
j.
2p
N 2( N-1)
j.
2p
N 3( N -1)
j.
2p
N
)
)
(e
j.
)4( N-1)
2p
N ...
)
ù
ú
ú é
ú ê
ú ê
ú ê
ú ê
ú ê
ú .ê
ú ê
ú ê
ú ê
ú ê
ú ê
ú êë
ú
ú
û
ù
X k(0) ú
X k(1) ú
ú
(2)
ú
Xk
ú
(3)
Xk ú
ú
X k(4) ú
ú
ú
( N -1)
ú
Xk
úû
= (N ) * IDFT -matrix
..and so can be realized by means of an N-point
`Inverse Discrete Fourier Transform’ (IDFT) !!!
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 19
OFDM Modulation
5/14
(0)
X
• PS: Note that k modulates a DC signal (hence often set to zero)
• PS: To ensure time-domain signal is real-valued, have to choose
*
( N-2)
(2) *
X(kN-1) = (X(1)
)
,
X
=
(X
) , ...
k
k
k
• PS: The IDFT matrix is a cool matrix:
– For any chosen dimension N, an IDFT matrix can be constructed as
given on the previous slide.
– Its inverse is the DFT matrix (symbol `F’).
DFT and IDFT matrices are unitary (up to a scalar), i.e.
F = (IDFT - matrix)-1 = N.(IDFT - matrix)H
– The structure of the IDFT matrix allows for a cheap (complexity
N.logN instead of N.N) algorithm to compute the matrix-vector
product on the previous slide (=IFFT =inverse fast Fourier transform)
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 20
OFDM Modulation
6/14
So this will be the basic modulation operation at the Tx :
Example: ‘16-QAM’
– The X’s are (QAM-symbols) defined by the input bit stream
Imag(X)
Real(X)
– The time-domain signal segments xk , xk+1, xk+2,..., xk+N-1
are
obtained by IDFT/IFFT and then transmitted over the channel, one
after the other. At the Rx, demodulation is done with an inverse
operation (i.e. DFT/FFT=fast Fourier transform).
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 21
OFDM Modulation
7/14
Sounds simple, but forgot one thing: channel H(z) !!
OFDM has an ingenious way of dealing with the channel effect,
namely through the insertion of a so-called `cyclic prefix’ at the Tx :
If the channel is FIR with order L (see p.10), then per segment,
instead of transmitting N samples, N+L sampes are transmitted
(assuming L<<N), where the last L samples are copied and put up
front…
L
x k  N  L ,..., x k  N 1
N
xk , xk+1, xk+2, xk+3,..., xk+N-L ,..., xk+N-1
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 22
OFDM Modulation
8/14
At the Rx, throw away L samples corresponding to cyclic
prefix, keep the other N samples, which correspond to
é
yk
ê
ê yk+1
ê
ê yk+2
ê yk+3
ê
ê yk+4
ê
ê
êë yk+N -1
ù é
ú ê
ú ê
ú ê
ú ê
ú=ê
ú ê
ú ê
ú ê
ú ê
úû êë
hL
hL-1
…
h1
h0
0
0
0
…
0
0
hL
…
h2
h1
h0
0
0
…
0
0
0
…
h3
h2
h1
h0
0
…
0
0
0
…
h4
h3
h2
h1
h0
…
0
0
0
…
h5
h4
h3
h2
h1
…
0
0
0
…
0
0
0
0
0
…
h0
é x
k+N -L
ê
ù ê xk+N -L+1
úê
úê
ú ê xk+N -1
úê
xk
ú. ê
úê
xk+1
úê
xk+2
úê
úê
xk+3
úû ê
ê
ê x
k+N -1
ë
ù
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
û
N+L
N
prefix
This is equivalent to …
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 23
OFDM Modulation
N
é
yk
ê
ê yk+1
ê
ê yk+2
ê yk+3
ê
ê yk+4
ê
ê
y
ê
ë k+N -1
ù
ú
ú
ú
ú
ú=
ú
ú
ú
ú
ú
û
é
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ë
0 ù
ú
0 ú
ú
0 úé
0 | I LxL
0 ú. ê
I NxN
úê
0 úë
ú
ú
h0 ú
û
hL
hL-1
…
h1
h0
0
0
0
…
0
hL
…
h2
h1
h0
0
0
…
0
0
…
h3
h2
h1
h0
0
…
0
0
…
h4
h3
h2
h1
h0
…
0
0
…
h5
h4
h3
h2
h1
…
0
0
…
0
0
0
0
0
…
h0
0
0
0
0
…
h3
h2
h1
h0
0
0
0
…
h4
h3
h2
h1
h0
0
0
…
h3
h2
h1
h0
0
…
h4
h3
h2
h1
h0
…
0
0
0
0
0
…
é
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ë
h4
h2
h1
9/14
ù
ú
ú
û
é
xk
ê
ê xk+1
ê
ê xk+2
. ê xk+3
ê
ê xk+4
ê
ê
x
ê
ë k+N -1
h1 ù
ú
h2 ú
ú
h3 ú
h4 ú
ú
ú
ú
ú
h0 ú
û
ù
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
û
N
(*)
The matrix (call it `H’) is now an NxN `circulant matrix’
=every row is the previous row up to a ‘cyclic shift’
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 24
OFDM Modulation
10/14
• PS: Cyclic prefix converts a (linear) convolution (see p.23)
into a so-called ‘circular convolution’ (see p.24)
• Circulant matrices are cool matrices…
A weird property (proof by Matlab!) is that when a circulant
matrix H is pre-/post-multiplied by the DFT/IDFT matrix, a
diagonal matrix is always obtained: FH.F-1 = [diagonal matrix]
Hence, a circulant matrix can always be written as
(=eigenvalue decomposition!)
é H
0 …
0
0
ê
0
H1 …
0
-1 ê
H = F .ê
ê
0 …
H N -1
êë 0
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
ù
ú
ú
ú.F
ú
úû
p. 25
OFDM Modulation
11/14
Combine previous formulas, to obtain…
é
yk
ê
yk+1
ê
ê
yk+2
ê
ê
yk+3
ê
yk+4
ê
ê
ê
yk+N -1
ê
ë
é
yk
ê
yk+1
ê
ê
yk+2
ê
F. ê
yk+3
ê
yk+4
ê
ê
ê
yk+N -1
ê
ë
ù
ú
ú
é H
ú
0
ê
ú
ú = F -1. ê 0
ê
ú
ê
ú
ê
ë 0
ú
ú
ú
û
ù
ú
ú
é H
ú
0
ê
ú
ê 0
ú=
ê
ú
ê
ú
ê
ë 0
ú
ú
ú
û
0
…
0
H1
…
0
0
…
H N -1
0
…
0
H1
…
0
0
…
H N -1
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
é
xk
ê
xk+1
ê
ù
ê
ú
xk+2
ê
ú
ê
xk+3
ú.F. ê
ú
xk+4
ê
ú
û
ê
ê
xk+N -1
ê
ë
é
xk
ê
xk+1
ê
ù
ê
ú
xk+2
ê
ú
ê
xk+3
ú.F. ê
ú
xk+4
ê
ú
û
ê
ê
xk+N -1
ê
ë
ù
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
û
ù
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
û
p. 26
OFDM Modulation
12/14
In other words…
é
(0 )
Y
k
ê
ê Y (1)
k
ê
ê Y (2 )
k
ê
ê Yk(3)
ê
( 4)
ê Yk
ê
ê
ê Yk( N -1)
êë
ù
é
ú
yk
ê
ú
ê yk+1
ú
ê
ú
yk+2
úD ê
ú = F. ê yk+3
ê
ú
ê yk+4
ú
ê
ú
ê
ú
êë yk+N -1
ú
úû
ù é H
0
ú ê
ú ê 0
ú ê
ú ê 0
ú=ê 0
ú ê
ú ê 0
ú ê
ú ê
úû êë 0
0
0
0
0
0
H1
0
0
0
0
0
H2
0
0
0
0
0
H3
0
0
0
0
0
H4
0
0
0
0
0
H N -1
é
(0 )
ù ê Xk
ú ê X (1)
k
úê
ú ê X (2 )
k
úê
ú. ê X (3)
k
úê
ú ê X k( 4)
úê
úê
úû ê X ( N -1)
k
êë
ù
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
úû
This means that after removing the prefix part and performing a DFT in the
Rx, the obtained samples Y are equal to the transmitted symbols X, up to
(scalar) channel attenuations Hn (!!)
Y
(n)
k
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
= Hn. X
(n)
k
p. 27
OFDM Modulation
13/14
• PS: It can be shown (check first column of F.H = [diagonal matrix].F ) that
Hn is the channel frequency response evaluated at the n-th carrier !
é H
0
ê
ê H1
ê
ê H2
ê H3
ê
ê
ê H N -1
ë
é h
0
ù
ê
ú
ê h1
ú
ê
ú
ê
ú = F. (1st column of H) = F. ê hL
ú
ê
ú
ê 0
ú
ê
ú
ê 0
û
ë
ù
ú
ú
ú
ú
ú
ú
ú
ú
ú
û
Þ
H n = H (z) z=e j 2 p n/ N
(p.27 then represents ‘frequency domain version’ of circular convolution,
i.e. ‘component-wise multiplication in the frequency domain’)
`Channel equalization’ may then be performed after the DFT (=in
by component-wise division (divide by Hn for carriern). This is referred to as `1-tap FEQ’ (Frequency-domain EQualization)
the frequency domain),
(n)
(n)
(n)
-1
(n )
YDSP-CIS
=
H
.
X
Þ
estimate{X
}
=
(H
)
.Y
/ Chapter-3:
n Acoustic
n
p. k
28
k
k Modem Project / Version 2012-2013
k
OFDM Modulation
14/14
• Conclusion: DMT-modulation with cyclic prefix leads to a
simple (trivial) channel equalization problem (!!)
CP insertion
CP removal
0
IFFT
P
/
S
Discrete
equivalent
channel
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
S
/
P
FFT
FEQ
p. 29
Target
Design efficient OFDM based modem (Tx/Rx)
for transmission over acoustic channel
Tx
D-to-A
A-to-D
Rx
Specifications:
Data rate (e.g. 1kbits/sec), bit error rate (e.g. 0.5%),
channel tracking speed, synchronisation, …
DSP-CIS / Chapter-3: Acoustic Modem Project / Version 2012-2013
p. 30