EE359 Discussion Session 9OFDM, Spread Spectrum, Multiuser Systems
December 6, 2017
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Outline
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Last discussion session
MIMO beamforming
Diversity multiplexing tradeoff
MIMO receiver design
Multicarrier modulation
This session
OFDM
Spread spectrum
Multiuser systems
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Outline
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Brief recap on multicarrier modulation (MCM)
Idea
Divide large bandwidth into smaller chunks and use narrowband signals
Pros and cons
Takes care of intersymbol interference (ISI)
Guard band versus spectral efficiency (SE)I No guard bands, high SE (e.g. OFDM): Overlapping subcarriers,
sensitive to timing/frequency offsetI Large guard bands, low SE (e.g. GSM FDM): Non overlapping
subcarriers, less stringent synchronization requirements
Multiplexing subcarriers
Analog: Use a separate modulator for each subcarrier and sum upsignals
Digital: Use a single modulator and then use a multiplexingarchitecture based on DFT (e.g. OFDM)
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Homework 8
Problem 1
Repetition coding makes an error if more than half of the received bits arein error. Moreover the error event is dominated by error events in channelswith low SNRs.
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Definition
DFT
Given N points x[0], . . . , x[N − 1], the DFT is given by
X[k] =
N−1∑i=0
x[i]e−j2πki/N =
N−1∑i=0
x[i]ωki, 0 ≤ k ≤ N − 1
Define Qω ∈ CN×N Qω,k,i = ωki 0 ≤ k, i < N , x ∈ CN s.t. xi = x[i],and X s.t. Xi = X[i]. If N is large, independent fading.
Some facts
X = Qωx
x = (Qω)−1X = Qω−1X =∑N−1
i=0 X[i]ω−ki (IDFT relation)
Matrix vector multiplication normally Θ(N2), but FFT/IFFTalgorithms allow Θ(N logN) complexity
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FDM block diagram
Σ s(t)
Modulator 0 D/A s0(t)
cos(2πf0t)
Modulator . . . D/A s...(t)
cos(2πf...t)
Modulator N − 1 D/A sN−1(t)
cos(2πfN−1t)
Demodulator i LPF and A/D r(t)
cos(2πfit)
Figure: FDM system with N subchannels
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OFDM block diagram
Modulator S/P IFFT CP, P/SCP, P/S D/A s(t)
cos(2πf0t)
X[n] x(t)
x0X0
x...X...
xN−1XN−1
Demodulator P/S FFT S/P, CP
removal
S/P, CP
removal
LPF and A/D r(t)
cos(2πf0t)
H[n]X[n] x(t)
x0H0X0
x...H...X...
xN−1HN−1XN−1
Figure: OFDM block diagram
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Need for cyclic prefix
Effect of channel
Channel does linear convolution, i.e.,
y[n] = h[n] ∗ x[n] + ν[n]
Not multiplicative under DFT, which needs circular convolution
Rationale for cyclic prefix (CP)
Circular convolution can be simulated for finite support h[n] byadding CP to x[n]
Math: y[n] = h[n] ~ x[n] =∑N−1
m=0 h[m]x[n−m], where x[n] =x[n mod N ]
If |h[n]| = 0 for n /∈ [0, µ], CP of length µ suffices to “simulate”circular convolution/remove ISI
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Final OFDM frame
X0 X1 X... XN−1
Frequency domain
xN−µ x... xN−1 x0 x... xN−1
Time domain (CP in red)
Some points
At the receiver simply discard the CP
One may also transmit null sequences instead of CP to “simulate”circular convolution (receiver processing different) (Problem 2,Homework 8)
Some features of OFDM based implementation
Efficient ISI removal ,
Inefficiency due to CP /
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Equivalent matrix representation of OFDM
Effect of channel (linear convolution)
y = Hx + ν where H is Toeplitz, yi = y[i], xi = x[i]
Effect of equivalent channel (circular convolution)
y = Hx + ν where H is circulant
Fact about circulant matrices
They can be diagonalized by the DFT matrix Qω!
OFDM can be thought of as transmit precoding and receiver shapingusing Qω and QH
ω
Homework 8
Problem 3 explores this connection
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OFDM Summary
Pros
Efficient ISI removal
Efficient implementation using FFT/IFFT
In a perfect system, subchannels are uncoupled
Cons
High peak to average power ratio
Sensitive to timing/frequency offset (which may couple adjacentsubchannels)
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Outline
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Introduction
Idea
Spread a narrowband signal over a wider band
Advantages
Signal not distinguishable from noise floor (military applications)
Narrowband interference rejection (again military applications)
Cellular resource allocation easier as compared to FDM/TDM
Disadvantages
Wideband receivers
Non orthogonality between spreading sequences (need equalization )
Two common incarnations
FHSS (Frequency hopping spread spectrum) (e.g. Bluetooth)
DSSS (Direct sequence spread spectrum) (e.g. CDMA in 3G )
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DSSS
Idea
At transmitter: Use a spreading code (also known as chip sequence)sc(t) of bandwidth B = 1/Tc (sometimes called chip rate) with whichto multiply narrowband signal g(t) of duration Ts = 1/Bc
At receiver: Take the integral of r(t)sc(t) over time Ts
Desirable properties for sc(t)
Autocorrelation function ρ(τ) , 1/Ts∫ Ts0 sc(t)sc(t− τ)dt = δ(τ):
gives multipath rejection
Orthogonality∫ Ts0 sc1(t)sc2(t)dt = 0 for c1 6= c2: inter user
inteference rejection
Practical properties for sc(t)
Given finite Ts and finite B, cannot have many orthogonal sequences
Design spreading codes with approximate orthogonality
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A specific sc(t)
Tc
Ts
1
-1
sc(t) in the time domain
1/Tc
Sc(f) frequency domain
At transmitter: Send s(t) = g(t)sc(t), g(t) narrowband
At receiver: Compute∫ Ts0 r(t)sc(t)dt
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Narrowband interference rejection properties
Narrowband interference
Let g1(t) be narrowband, then system model is r(t) = s(t) + g1(t)
Spreading action at receiver
r(t)sc(t) = g(t) + g1(t)sc(t)
1/Tc
1/Ts
Figure: Frequency dom. of r(t)sc(t). Integration ≡ lowpass filtering
Narrowband interference rejection
Thus g1(t) is reduced by a factor of approximately Ts/Tc (spreading factor)
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Homework 8
Problems 4
Explore the statistics of interference and noise in the spread spectrumsystem
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Multipath (ISI) rejection properties
Multipath
Let h(t) = δ(t) + αδ(t− τ), then r(t) = s(t) + αs(t− τ)
Action at receiver
1/Ts∫ Ts0 r(t)sc(t)dt = gρ(0) + α/Tsρ(τ)
Autocorrelation function ρ(τ) of sc(t)
Smaller the support of ρ(τ), better is multipath/ISI rejection
Larger the support of ρ(τ), better is synchronization at receiver
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Rake receivers
Idea
Using spreading codes with good ISI rejection, we can distinguish differentmultipath components!
Some facts
RAKE receiver simply gathers energy from different multipathcomponents with different delays
Different branches of the RAKE receiver synched to a different delaycomponent
Can be combined using diversity combining techniques (MRC/SC,etc.)
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Homework 8
Problem 5
Use RAKE receivers with different diversity combining schemes
Problem 6
Sharing frequency, time or code resources for multi-user communications
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Outline
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Multiuser Communications
Centralized schemes - Resource divided in frequency, time, or codespace and allotted to users. (Cellular systems)
Decentralized schemes - Users attempt to communicate throughrandom access, if there are no collisions, it is successful. Else,transmit with random delay.
Cellular systems - Area divided into cells, different frequencies (reuse)could be assigned to them.
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Homework 8
Problem 7
In centralized control, there are no collisions. For part (b), writing theindicator function that user i transmits in slot j would help - takingexpectations of these functions would throughput.
Problem 8
Frequency divided in half along adjacent cells. Interference is calculatedfrom closest cells.
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