ECE 5884/6884 Wireless Communications
Week 8 Lecture
Wireless Channel Equalization
techniques for Single Carrier
Copyright By PowCoder代写 加微信 powcoder
and Multicarrier Systems (OFDM)
Dr. Sessional Lecturer
Single Carrier and Multicarrier
Single Carrier systems
• Time domain equalization
• Frequency domain equalization
Multicarrier systems
• (Orthogonal Frequency Division Multiplexing OFDM) + Frequency domain equalization
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Single Carrier Systems
Least squares time domain equalizer for single carrier systems
• Learning objective: Develop Least Squares based channel equalizer to compensate for the effect of wireless channel induced inter symbol interference (ISI)
Channel Estimate based time domain equalization (TDE)
Removing ISI using linear equalization
• Consider an FIR linear equalizer with coefficients for
• Write as a linear system
Toeplitz structure
Lf +L+1 X Lf +1 (tall)
Computation of the LS equalizer
• Toeplitz structure in H leads to efficient algorithms to solve LS • H is full rank as long as at least one coefficient is nonzero
• The LS solution assuming H is full rank is
• with squared error
• The squared error can be further minimized by choosing nd
Example: ISI channel
4-PAM s[n]
MSE function of filter delay
LS equalizer, order 3
optimal delay
MSE function of filter delay
Example: LS equalizer, order 10
optimal delay
^ 4_PAM s[n]
Receiver with channel estimation and linear equalization
LMMSE or LS computation
LMMSE estimator
• Suppose we want to estimate x from an observation y
• Unknown vector x of size M x 1 with zero mean and covariance Cxx • Observation vector y with zero mean and covariance Cyy
• x and y are jointly correlated with covariance matrix Cyx
• The objective of the LMMSE estimator is to determine a linear transformation such that
• Equivalently
LMMSE estimator
• Solving for one column of GMMSE
• MMSE orthogonality equation
• Taking the expectation and setting the result to zero
• Solution is
LMMSE estimator
• Reassembling the column of G and combining the results together
• The MMSE estimate of x is
Reformulating the equalization problem
• Consider the received signal after the equalizer where
𝑠 𝑛 =[𝑠 𝑛 ,𝑠 𝑛−1 ,…𝑠 𝑛−𝐿 ,0,0,0…]
• The LMMSE equalizer minimizes the error 15
𝐿+𝐿+1𝑋𝐿+1 𝑓𝑓
Solving for the LMMSE equalizer
• Assuming that
• s[n] is IID with zero mean and unit variance, • v[n] is IID with variance
• s[n] and v[n] are independent
• Then the estimation mean squared error is given by •
• The LMMSE equalizer is computed 16
Linear equalization discussion
• Connections between LMMSE and LS solutions noise variance
becomes large
matched filter
• Complexity of the equalizer depends on the choice of Lf
• Larger values give better performance but higher complexity
• Lf a design parameter that is generally greater than the ISI length L
• As time domain equalizers need Lf L complexity grows with the channel introduced ISI
noise variance goes to zero
LS solution
Linear Time domain equalization
LMMSE or LS computation
Linear time domain equalizer (summary)
• TDE for single carrier systems is calculated using the LS or LMMSE optimization criterion.
• Asymptotically when 2 → 0 LMMSE approaches LS equalizer v
• SNR estimation is not required for LS approach, hence less complex.
• Computational complexity of Linear-TDE depends on L and Lf
• The choice of the equalizer length affects the delay nd.
• Computational complexity of TDE is 2 , tolerable for small equalizer 𝑂(𝐿𝑓)
lengths but not practical for longer equalizers.
Multipath effect and Equalizer length (TDE)
• Multipath propagation induces ISI (measured in number of symbol periods 𝐿) in the time domain. As a consequence of ISI, frequency domain characteristics exhibit frequency selectivity.
• Example, a channel with a maximum delay spread 10μsec on the transmission of symbol rate 𝑅𝑠 introduces
• ISIof𝐿=10ata𝑅𝑠 =1𝑀
• ISI of 𝐿 = 100 at a 𝑅𝑠 =10𝑀
• ISIof𝐿=1000ata𝑅𝑠 =100𝑀 • ISI of 𝐿 = 10000 at a 𝑅𝑠 =1𝐺
Is it practical to implement a TDE for transmisión links that support high data rates over multipath (non-line of sight ) fading channels?
Frequency Domain Equalization
• TDE require a convolution on the received signal to remove the effects of the channel.
• It is desirable to have receiver complexity that does not change with the channel delay spread.
• An alternative to TDE is to perform equalization completely in the frequency domain.
• Frequency Domain Equalization (FDE) is implemented as component wise multiplication.
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Frequency domain equalization
Frequency domain equalization (FDE) is channel compensation in the frequency domain. FDE is based on the discrete Fourier transform (DFT) of signals
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Circular Shift Property of DFT
Periodic for the memory of the channel
Component wise multiplication in the frequency domain is convolution in the time domain
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Circular convolution and Linear Convolution
Unfortunately, linear convolution, not circular convolution, is a good model for the effects of wireless propagation.
It is possible to mimic the effects of circular convolution by modifying the transmitted signal with a suitably chosen guard interval. The most common choice is what is called a cyclic prefix,
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Single Carrier – Cyclic Prefix
Use cyclic prefix to convert a linear convolution to a circular convolution and equalize in the frequency domain. Convolution of signal with channel should appear circular.
Last 𝐿𝑐 QAM symbols appended at the start of transmission
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Single Carrier – Cyclic Prefix
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
• Organize data into blocks of N symbols
• Assemble a block of N + Lc symbols where • Prefixed part
• Data part
• Neglect the first Lc terms of the convolution
Cyclic prefix
Exposing the circular convolution 3/3
Connection to frequency domain equalization
• Conclude that the received signal is equivalently
• Using the DFT equalization can proceed as
• Can use frequency domain equalization! 29
𝑤[𝑛] 𝑠[𝑁−𝐿𝐶] 𝑠[𝑁−1]
SC-FDE transmitter
Frame sync determines the start of the cyclic prefix
SC-FDE receiver
creates the blocks
TDE vs FDE
• In general, the linear time domain equalizer (TDE) length 𝐿 ≥ 𝐿 𝑓
• FDE of FFT size 𝑁 >> 𝐿 offers complexity savings over the TDE for large 𝐿.
• As 𝑁 >> 𝐿 it increases latency at the receiver for FDE systems.
• An important requirement for FDE is circular transmisión such as a
cyclic prefix (CP). The CP overhead makes TDE throughput inefficient
compared to the FDE approach.
• More complexity savings can be realized in a MIMO scenario.
Multicarrier systems
Why is OFDM so successful?
• Growing demand for increased data rates
• Overall complexity of OFDM is practical on channels with long delay spreads.
• OFDM allows the signal to be tailored to the channel. Large constellations on frequencies (subcarriers) with high signal to noise ratio (SNR)
• Forms the basis of OFDMA (Orthogonal frequency division multiple access)
• OFDMA is used instead of TDMA, FDMA or CDMA in the LTE fourth
generation mobile systems
• Provides spectrum flexibility for example in future cognitive radio systems
What is OFDM?
• OFDM is a particular form of multicarrier system
• uses discrete Fourier Transform/Fast Fourier Transform
• sin(x)/x spectra for subcarriers
• Available bandwidth is divided into narrow bands (subcarriers)
• ~2000-8000 for digital TV • ~48 for Hiperlan 2
• Data is transmitted in parallel on these bands
Multicarrier systems
Single carrier system
signal representing each bit uses all of the available spectrum
Multicarrier system
available spectrum divided into many narrow bands
data is divided into parallel data streams each transmitted on a separate band
IDFT at the transmitter
Interference free communications?
Each subcarrier has a different frequency
Frequencies chosen so that an integral number of cycles in a symbol period
Signals(sub-carriers) are mathematically orthogonal
T 2kt −2lt
sin T sin T dt=0, kl
Cosine components of first three subcarriers
Symbol period
T 2kt −2lt
sin T cos T dt=0, allkandl 0
Key is orthogonality of subcarriers
Frequency W/N 38
How is data carried on the subcarriers?
• Data is carried by varying the phase or amplitude of each subcarrier
• Quadrature phase shift keying (QPSK), Quadrature Amplitude Modulation (QAM), 4-QAM, 16-QAM, 64-QAM
• The overall OFDM symbol is then formed from the sum of
all orthogonal subcarriers.
Two possible subcarrier values
(Cosine component only)
• The overall OFDM symbol has a finite duration T, hence the effect of windowing in the time domain leading to Sinc shaped spectra of subcarriers.
Frequency W/N 39
Baseband OFDM system
Discrete frequency domain Each input controls
signal at one frequency
High speed data
Discrete Time Domain Samples of modulated and multiplexed signals
Cyclic Prefix
+ Parallel to Serial
High speed data
Error Decoding
+ Parallel To Serial
Serial to Parallel + Error coding
Equal- izer
IFFT and FFT are the main components in the transmitter and receiver
Parallel Remove C P
R F Filter A/D etc
Low Pass Filter
Baseband OFDM system
Discrete frequency domain Each input controls
signal at one frequency
Discrete Time Domain Samples of modulated and multiplexed signals
Cyclic Prefix
+ Parallel to Serial
High speed data
Serial to Parallel + Error coding
D/A Filter RF
a exp k
Samples at IFFT Samples at IFFT output input
Reference from ECE 4024 by Prof.
How are OFDM signals generated?
• Parallel data streams are used as inputs to an IFFT
• IFFT output is sum of signal samples
• IFFT does modulation and multiplexing in one step
• Filtering and D/A of samples results in baseband signal
Signal values at the output of the IFFT are the sum of many samples of many sinusoids – looks random
Reference from ECE 4024 by Prof.
Typical IFFT Output Samples
Signals at the input/output of the transmitter IFFT
OFDM Frequency Domain Symbol
-10 5 10 15
Subcarrier Index m
-10 5 10 15
Subcarrier Index m
Complex numbers representing data to be
transmitted in this OFDM symbol
OFDM Time Domain Signal Samples
Discrete Time Index k
N−1 1 j2kl
b = a exp k
Reference from ECE 4024 by Prof.
lNk=0 N 16 subcarriers
Discrete Time Index k
Complex numbers representing samples of signal to be transmitted
imag(X) real(X)
imag(x) real(x)
Signals at the input/output of the transmitter IFFT
OFDM Frequency Domain Symbol
-10 5 10 15
Subcarrier Index m
-10 5 10 15
Subcarrier Index m 4-QAM constellation
2 1.5 1 0.5 0 -0.5 -1
N =16 16 subcarriers
OFDM Time Domain Signal Samples
Discrete Time Index k
-1.5 Reference from ECE 4024 by Prof. -2
Discrete Time Index k
Output has Gaussian distribution for N >64
-2 -1 0 1 2 real(X)
imag(X) real(X)
imag(x) real(x)
Signals at the input/output of the IFFT, N=32
OFDM Frequency Domain Symbol
0 5 10 15 20 25 30
-2121201200 5 D10Sisucbrectaer1rTi5eimr IendIne2dx0emx k 25 30
Subcarrier Index m
OFDM Time Domain Signal Samples
0 5 10 15 20 25 30
Discrete Time Index k
0 5 10 15 20 25 30
Subcarrier Index m 16-QAM constellation
2 1.5 1 0.5 0 -0.5 -1 -1.5
-2 -1 0 1 2
N = 32 32 subcarriers
0 5 10 15 20 25 30
Discrete Time Index k
Reference from ECE 4024 by Prof. 45
Output has Gaussian distribution for N >64
imag(X) real(X)
iimag(Xx ) reall(Xx )
OFDM Receiver
High Error
speed Decoding Equal-
Parallel Remove CP
RF Filter A/D etc
+ izer Parallel
• Key component is the discrete Fourier transform (DFT/FFT)
• It demultiplexes the subcarriers and demodulates them
Reference from ECE 4024 by Prof.
Transmitter and receiver
Each block in the receiver performs the ‘inverse’ of the corresponding transmitter function
Ideally what goes in at the transmitter comes out at the receiver
High speed data
Serial to Parallel + Error coding
Cyclic Prefix
+ Parallel to Serial
Low Pass Filter
High speed data
Error Decoding
+ Parallel To Serial
Reference from ECE 4024 by Prof.
Equal- izer
Parallel Remove CP
RF Filter A/D etc
Received Constellation – 4-QAM
No distortion in channel
2 1.5 1 0.5
-0.5 -1 -1.5 -2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 real(X)
0.5 0 -0.5 -1 -1.5 -2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 real(Y)
Reference from ECE 4024 by Prof.
-0.5 -1 -1.5 -2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 real(Y)
Increasing Noise levels (decreasing SNR)
The Transmitter IFFT in more detail
High speed data
Outputs when only one input is non zero and that input = 1 (real only)
Serial to Parallel + Error coding
Cyclic Prefix
+ Parallel to Serial
0.1 0.1 0.1 000
0 5 10 15 20 25 30
Discrete Time Index m
0 5 10 15 20 25 30
Discrete Time Index m
0 5 10 15 20 25 30
Discrete Time Index m
0 5 10 15 20 25 30
Discrete Time Index m
Second Subcarrier
0 5 10 15 20 25 30
Discrete Time Index m
Zeroth Subcarrier
0 5 10 15 20 25 30
Discrete Time Index m
First Subcarrier
Reference from ECE 4024 by Prof.
Low Pass Filter
imag(x) real(x)
imag(x) real(x)
imag(x) real(x)
OFDM in a multipath environment effect on one subcarrier
In Multipath propagation envirornment,
Received signal in one symbol period is not a sinusoid Causes intercarrier interference (ICI)
First symbol
Second symbol
Signal on Path 1
Signal on Path 2
Signal on Path 1+2
ICI Reference from ECE 4024 by Prof.
Cyclic Prefix
Each symbol is cyclically extended
Some loss in efficiency as cyclic prefix carries no new information
Cyclic Prefix
Symbol without prefix
Cyclic Prefix
Signal transmitted on one subcarrier for one symbol
Reference from ECE 4024 by Prof.
Effect of multipath on symbol with cyclic prefix
If multipath delay is less than the cyclic prefix
no intersymbol or intercarrier interference amplitude may increase or decrease
Signal on Path 1 Signal on Path 2
Path delay
Cyclic Prefix
Reference from ECE 4024 by Prof.
Frequency selective fading
-10 0.5 1 1.5 2
SymbolDuration
-20 0.5 1 1.5 2 SymbolDuration
Transmitted Signal
Main signal + Delayed signal
Transmitted Signal
Main signal + Delayed signal
Reference from ECE 4024 by Prof.
Amplitude Amplitude
Spectrum of an individual OFDM subcarrier
convolve delta
frequency domain
sin(x)=sinc(x) x
Shape of spectrum depends on shape of window
Width of spectrum depends on time duration of window
Centre frequency of spectrum depends on frequency of sinusoid
Time domain subcarrier
subcarrier 𝑓 𝑘
Frequency domain
Time domain
Reference from ECE 4024 by Prof.
Spectrum of Received Signal
Multipath fading causes some frequencies to be attenuated Fading is approximately constant over narrow band (sub-channel) This is corrected in the receiver
Reference from ECE 4024 by Prof.
Amplitude and phase change complex baseband equivalent channel
• Multipath delay causes change in amplitude and phase of each subcarrier
• Change depends on subcarrier frequency
• Can consider the channel ‘seen’ by the k’th subcarrier as a complex equivalent baseband channel with frequency response 𝐻
• Corrected in receiver by one complex multiplication per subcarrier
• Multiply by 1
Reference from ECE 4024 by Prof.
Complex baseband equivalent of k’th subcarrier 56
-10 0.5 1 1.5 2
SymbolDuration
-20 0.5 1 1.5 2 SymbolDuration
Transmitted Signal
Main signal + Delayed signal
Transmitted Signal
Main signal + Delayed signal
𝑌=𝐻 𝑋 +𝑁 𝑘𝑘𝑘𝑘
Amplitude Amplitude
OFDM with N sub carriers
This convolution with delta function at 𝑓 results in the frequency 0
shift of the Sinc pulse to 𝑓 0
OFDM spectrum
𝜟𝒇 Subchannel spacing IFFT
Spectrum – Effect of Number of subcarriers, N (no cyclic prefix)
N=64, cp=0
N=128, cp=0
-5 -10 -15 -20 -25 -30 -35 -40
N=32, cp=0
-5 -10 -15 -20 -25 -30 -35 -40
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Normalized Frequency
-5 -10 -15 -20 -25 -30 -35 -40
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Normalized Frequency
N = 128 N=128, cp=0
OFDM SPECTRUM
Fourier transform of
individual subcarriers
0.8 0.6 0.4 0.2
0.8 0.6 0.4 0.2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Normalized Frequency
N = 32 111
N=32, cp=0
N=64, cp=0
000 -0.2 -0.2 -0.2 -0.4 -0.4 -0.4 -0.6 -0.6 -0.6
More subcarriers = Faster Spectral Roll-off
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
Normalized Frequency Normalized Frequency Normalized Frequency
Reference from ECE 4024 by Prof.
Continuous Fourier Transform of 0-th Subcarrier
Spectrum of OFDM signal (dB)
Continuous Fourier Transform of 0-th Subcarrier
Spectrum of OFDM signal (dB)
Continuous Fourier Transform of 0-th Subcarrier
Spectrum of OFDM signal (dB)
Histograms with N
The Receiver Equalizer
High speed data
Error Decoding
+ Parallel To Serial
Equal- izer
Multipath fading affects the gain and phase of each subcarrier The equalizer corrects the gain and phase of each subcarrier
Reference from ECE 4024 by Prof.
Parallel Remove CP
RF Filter A/D etc
Spectrum of Received Signal
Equalization using Single Tap Equalizer
Parallel + Remove CP
Parallel to Serial
Decod- ing
High speed data
‘Single Tap Equalizer’ Equalization achieved by a single (complex) multiplication per subcarrier Complexity increases linearly with data rate
Reference from ECE 4024 by Prof.
Equalization using Sin
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