ECE 5884/6884 Wireless Communications Week 7 Lecture
Pulseshaping and Matched Filtering Synchronization
Channel Estimation
Dr. Sessional Lecturer
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Discrete-time processing of continuous-time signals
Objective: Determine equivalent combination of sampling, digital filtering, and reconstruction to process a bandlimited continuous-time signal with discrete-time signal
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
processing
Nyquist sampling theorem
Let 𝑥 (𝑡) be a bandlimited signal, which means that 𝑋 (𝑓) = 0 for 𝑓 ≥ 𝑓 . Then 𝑥 (𝑡) is 𝑐𝑐𝑁𝑐
uniquely determined by its samples {𝑥[𝑛] = 𝑥 (𝑛𝑇 )} n ∈ [−∞, ∞] 𝑐𝑠
If the sampling frequency satisfies
𝑓 ≔ 1 ≥ 2𝑓 𝑠𝑇𝑁
where 𝑓 is the Nyquist frequency and 2𝑓 is generally known as the Nyquist rate.
𝑥𝑐 (𝑡) = 𝑛
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
visualize the sampling operation
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
The effect of sampling on the signal bandwidth
DAC and ADC
Generation of discrete to continuous waveform at the transmitter-practically implemented using the digital to analog converter (DAC)
Generation of continuous to discrete samples at the receiver -practically implemented using the analog to digital converter (ADC)
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Discrete-time equivalent channel
A continuous time domain signal at the receiver 𝑦 𝑡 is sampled to obtain its 𝑐
discrete-time equivalent for carrying out digital signal processing
Low-pass filtered continuous-time channel
Discrete-time equivalent channel
Sampled Tx baseband signal
Baseband Rx signal samples
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Discrete-time channel
Pulse shaping
What is the best pulse shape for transmission over Wireless channel?
Baseband Transmitter
Baseband Receiver
Symbol Mapping
Up- sampling
Pulse Shaping Filter
Baseband Channel
Symbol de-mapping
Down Sampling
Matched Filter
h 𝑡=𝑔𝑡⊗𝑔(𝑡)⊗h(t) 𝑒𝑓𝑓 𝑡𝑥 𝑟𝑥
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Pulse shaping filter design criteria
Ideally in the continuous time domain,
the effective pulse shape 𝑔 𝑡 = 𝑔 𝑡 ⊗ 𝑔
In the discrete-time domain 𝑡 = 𝑛𝑇, 𝑔𝑛𝑇=𝑔 𝑛𝑇⊗𝑔 𝑛𝑇=𝛿(𝑛𝑇)
So that the Fourier transform of the sampled 𝑔 𝑛𝑇 is
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
𝐺 𝑓+𝑘 =𝑇 𝑇
Nyquist criterion for Pulse shaping
The continuous-time received signal at the baseband corresponding to transmit signal 𝑠(𝑚)
𝑦𝑡=𝐸h 𝑡⊗σ𝑠𝑚𝛿𝑡−𝑚𝑇+𝑔(𝑡)⊗𝑣(𝑡) 𝑥𝑒𝑓𝑓𝑚 𝑟𝑥
The discrete-time received signal at the baseband corresponding to transmit signal 𝑠(𝑚)
𝑦𝑛𝑇 = 𝐸 h 𝑛𝑇 ⊗σ 𝑠𝑚𝛿𝑛𝑇−𝑚𝑇 +𝑔 (𝑛𝑇)⊗𝑣(𝑛𝑇) 𝑥𝑒𝑓𝑓 𝑚 f 𝑟𝑥
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Discrete-time Received Signal
Assume the baseband channel h(t) = 1 Then, h𝑒𝑓𝑓(𝑡) = 𝑔𝑡𝑥(𝑡) ⊗ 𝑔𝑟𝑥(𝑡)
𝑦𝑛𝑇 = 𝐸 𝑠𝑚𝑔𝑛𝑇−𝑚𝑇 +𝑔 (𝑛𝑇)⊗𝑣(𝑛𝑇) 𝑥 𝑟𝑥
𝑦[𝑛]= 𝐸 𝑠𝑚𝑔[𝑛−𝑚]+𝑔 [𝑛]⊗𝑣[𝑛] 𝑥 𝑟𝑥
Signal component Noise component
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Zero-ISI Criterion
Signal Energy is calculated at the sampling instant i:e 𝑚 = 𝑛 Ε 𝐸𝑠𝑛𝑔[0]2
Energy at all other sampling instants i:e 𝑚 ≠ 𝑛 interferes with the detection of other
symbols and is termed as Inter-Symbol-Interference (ISI)
Ε σ 𝐸 𝑠 𝑚 𝑔[𝑛 − 𝑚] 2 𝑚≠𝑛 𝑥
σ𝐸 𝑔(𝑚𝑇)2 𝑚 =⋯−1,0,1…. 𝑥
Design Goal is to satisfy the Zero-ISI Criterion for the pulse
𝑔𝑛𝑇 =𝑐𝛿(𝑛)
σ𝐸 𝑔[𝑚𝑇]2=0 𝑥
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
𝑔𝑡=𝑔𝑡𝑥 𝑡 ⊗𝑔𝑟𝑥 −𝑡
What are the pulses that satisfy Nyquist criterion?
• The standard sync pulse satisfies the Nyquist criterion, however, the impulse response
1. Is a non-causal system (with impulse response non-zero for t<0) and hence difficult to approximate
2. The slope of the sync waveform is 1/t at time instants other than zero crossings which is very slow.
3. Due to this it is very sensitive to sample timing errors causing significant interference to adjacent symbols.
4. faster decay such as 1 or even 1 are desirable to minimize the ISI due to timing jitter in adjacent samples 𝑡2 𝑡3
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Design of desired pulse shapes using Nyquist criterion
◆ In the frequency domain,
G(f)= f Z(f)
where, Z(f)=Z(−f) and,Z(f)=0, ffs1
𝑔𝑟𝑐 = sin 𝜋 𝑡Τ𝑇 𝑧(𝑡) 𝜋𝑡Τ𝑇
Even Function Band limited filter
◆ In the time domain,
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
𝑔𝑟𝑐 (𝑡) = sin 𝜋 𝑡Τ𝑇 𝜋𝑡Τ𝑇
cos(𝜋𝛼𝑡) 1− 2𝜋𝛼𝑡Τ𝑇
Raised Cosine Pulse
𝑇, 𝐺𝑓=𝑇1+𝑐𝑜𝑠𝜋𝑇𝑓−1−𝛼 ,
0≤ 𝑓 ≤1−𝛼 passband 2𝑇
2𝑇 ≤ 𝑓 ≤ 2𝑇 transition band
𝑓 >1+𝛼 outofband 2𝑇
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Impulse response of the Raised Cosine filter
Questions on impulse response of the RC pulse shape
1. What is the effect of filter span (sometimes referred to as fil ter order) on the transmitted signal?
2. What are the implications of choosing a long filter span?
3. What does roll-off factor α in pulse shaping filters control?
4. What are the implications of designing pulse shaping filters with 0<α<1.
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Frequency response of the Raised Cosine filter
Questions on frequency response of the RC pulse:
1. What is the effect of filter span (sometimes referred t o as filter order) on the spectrum of the transmitted si gnal?
2. Is there an ideal choice for the filter span?
3. How does roll-off factor α in pulse shaping filters affec t the pass band and out of band of the transmit signal ?
4. What are the implications of designing a pulse shapin g filters with 0<α<1.
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Eye Diagrams
• The width of the eye opening defines the time interval over which the received signal can be sampled without error from ISI. It is intuitive that the preferred time for sampling is the instant of time at which the eye is open the widest.
• Sensitivity of the system to timing errors is determined by the slope of the eye as the sampling time is varied.
• The height of the eye opening specifies the noise margin of the system
• Pulses with more distortion of zero-crossings imply susceptibility to synchronisation errors.
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Comparison of filters with different alphas
RRC pulse shape with 10% roll-off
RRC pulse shape with 100% roll-off
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Channel impairments
• Time offset
• Frequency offset
• Multipath channel distortions
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Timing offset
Transmit Frame
Received Frame
Received Frame
Received Frame
Received Fram
Rx Frame Window
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Time Synchronization
1. How to know the start of the Frame?
✓ Frame Synchronization
2. How to get the optimal sampling instants from the oversampled received signal?
✓ Symbol synchronization
Reference: Introduction to Wireless Digital Communications: A Signal Processing Perspective
Timing synchronization algorithm
Two-stage Timing synchronization algorithm
Let, the propagation time in sec be denoted by
𝜏 =𝜏𝑖𝑛𝑡𝑒𝑔𝑒𝑟+𝜏𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝜏𝑑=𝑑𝑇+𝑘∗𝑇
𝒅isdelayinnumberofsymbolperiods,and𝒌∗is the optimal fractional delay sample periods
Correlation based Peak detection for frame synchronization.
Energy based symbol synchronization Performed on over-sampled signal
Output energy tracking
Consider transmission of a Nyquist pulse shaped signal with a waveform as shown in the figure below
Store energy from current sampling Store energy from previous sampling
Output Energy Maximization
Symbol Synchronization
𝑦 [ 𝑛 ] = 𝑟 𝑚 𝑔 𝑟 𝑥 𝑛 − 𝑚 𝑚=−∞
We use this sampled signal to compute a discrete-time version of JMOE(τ) given by
𝐽 [ 𝑘 ] = Ε [ 𝑦 𝑛 𝑀 + 𝑘 2 ] 𝑀𝑂𝐸
where k is the sample offset between 0, 1, . . . , M − 1 corresponding to an estimate of the fractional part of the timing offset given by kT/M
replace the expectation with a time average over P symbols
𝐽 [ 𝑘 ] = 𝑀𝑂𝐸
𝑦 [ 𝑝 𝑀 + 𝑘 ] 𝑝=0
𝑘 is the is the sample delay offset
that maximizes the 𝐽 𝑀𝑂𝐸
Frame Synchronization
The objective is to determine the transmission delay 𝑑 𝑦[𝑛] = 𝐸 𝛼𝑒𝑗𝜙𝑠[𝑛 − 𝑑] + 𝑣 𝑛 𝑥
Transmission frame with a training sequence 𝑁 appended at the start 𝑡
A correlation based detector correlates the received with the training sequence to
𝑅[𝑛] = 𝑡∗ 𝑝 𝑦[𝑛 + 𝑝 ] 𝑝=0
And, an estimate of the delay is obtained as the index n that corresponds to maximum
correlator output
መ 𝜏𝑑 = 𝜏𝑖𝑛𝑡𝑒𝑔𝑒𝑟 + 𝜏𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 26 𝑑=max𝑛 𝑅[𝑛] 𝑑 𝑑
Frequency Offset
o What is the origin of frequency offset?
o Analyze a simple frequency offset estimation algorithm based on sending training sequence
Downconversion
• Consider the received signal after downconversion
• What if only not fc is known at the receiver? • The result is carrier frequency offset (CFO)
Carrier frequency offset (CFO)
CFO in Hz CFO normalised
Question: A certain digital transmission scheme has 1 M symbols per sec and a CFO of 200Hz. What is the normalized CFO?
The effect of frequency offset on discrete-time signal
• Assume the offset is small, the front-end bandwidth is sufficiently wide, 𝑦 𝑡 = 𝑒 𝑗2𝜋𝑓 𝑡 (h 𝑡 ⊗ 𝑠 𝑡 ) + 𝑣(𝑡)
𝑦[𝑛]=𝑒𝑗2𝜋𝜖𝑛(h𝑛 ⊗𝑠𝑛)+𝑣(𝑛)
• Rotation occurs after the convolution
• Impacts channel estimation and thus equalization
• In discrete time(𝑡 = 𝑛𝑇), including noise and with 𝜖 = 𝑓 𝑇 𝑒
h 𝑡=𝑔𝑡⊗𝑔(𝑡)⊗h(𝑡)=h(𝑡) • Assume, a Matched filter implementation so that, 𝑒𝑓𝑓 𝑡𝑥 𝑟𝑥
Visualizing the frequency offset effect
• Special case of flat fading channel 𝑦[𝑛]=𝑒𝑗2𝜋𝜖𝑛h𝑛𝑠𝑛 +𝑣(𝑛)
• 𝜖 is generally small but unknown
• Rotates constellation by 𝑒
Frequency offset synchronization
• Frequency offset is a severe impairment
• Even a small offset leads to significant degradation
• Frequency offset synchronization is challenging
• Offset occurs after the convolution with the unknown channel • Impacts channel estimation and frame synchronization
• Methods for offset correction
• Exploit structure in the received signal
• Create and exploit structure using a known training signal
Frequency offset estimation and channel estimation
Assume transmission of two blocks of 𝑁 training symbols ( ) each 𝑡
𝑠𝑛=𝑠𝑛+𝑁 =𝑡𝑛 𝑓𝑜𝑟 𝑛=0,1,....𝑁−1 𝑡𝑡
This transmission structure is exploited at the receiver to estimate The received sequence at the sampling instant 𝑛 is
𝑦[𝑛] = 𝑒 𝑗2𝜋𝜖𝑛 (h[𝑛] ⊗ 𝑠[𝑛]) + 𝑣[𝑛]
The received sequence at the sampling instant 𝑛 + 𝑁 is 𝑡
𝑡 𝑡 𝑡 33 = 𝑒 𝑗2𝜋𝜖(𝑛+𝑁 ) h[𝑛]⊗𝑠[𝑛+𝑁 ] +𝑣[𝑛+𝑁 ]
Frequency synchronization using training symbols
Exploiting the training structure of the frame format
𝑠𝑛=𝑠𝑛+𝑁 =𝑡𝑛𝑛=0,1,....𝑁−1 𝑡𝑡
The discrete time received signal at 𝑛 and 𝑛 + 𝑁 is given by 𝑡
𝑡𝑡 𝑦𝑛+𝑁 ≈𝑒𝑗2𝜋𝜖𝑁 𝑦𝑛
𝑦𝑛+𝑁 ≈𝑎𝑦𝑛 𝑡
The goal is to estimate a and apply to the entire transmission frame, hence the objective function is
𝐽𝑎 = 𝑦𝑛+𝑁 −𝑎𝑦𝑛 𝑡
a l=L (y[n+N ]−ay[n]) y[n+N ]=0
aˆ = l = L
y[n]y[n+N ] t
Frequency synchronization using training symbols
| y[n+N ]−ay[n]|2= t
(y[n+N ]−ay[n]) (y[n+N ]−ay[n])=0
Applying orthogonality principle
y [n+N ]y[n+N ]
The normalised frequency offset is obtained from the angle
aˆ = t
y [ n ] y [ n + N ] t
y[n]y[n+N] t
y[n]y[n+N ] t
Is the estimate of the normalised frequency offset
Effect of frequency offset on digital constellation
CFO estimation error decrease from 5% to 0.1%
Channel estimation and Equalization
𝑦[𝑛]=𝑒𝑗2𝜋𝜖𝑛(h𝑛 ⊗𝑠𝑛)+𝑣(𝑛)
Next task is channel estimation and equalization
Receiver Processing stages
Synchronization
Time domain Equalization
Data detection
Training based Channel estimation
1. Channel in general is assumed to be causal and finite impulse response (FIR).
1. Each multipath component arrives with a different delay and phase shift
2. More channel parameters for estimation in the multipath/frequency sel
ective channels when compared to frequency –flat channels.
Model for the received signal
• Consider the received signal after matched filtering and sampling
Channel distorted signal
Transmitter output
• The effective channel
Model for the received signal
• Samples obtained at the output of receive matched filter and
down sampler
𝑦 𝑛 = √𝐸 h 𝑙 𝑠 𝑛 − 𝑙 + 𝑣[𝑛] 𝑥
• Suppose the channel is FIR and the signal decays with distance, meaning long reflections are very weak. The simplified system with FIR
Order of the FIR channnel, usually assumed known
𝑦𝑛 =√𝐸 h𝑙𝑠𝑛−𝑙 +𝑣[𝑛] 𝑥
• Suppose that
• This is a two-tap discrete-time channel
Inter Symbol Interference (ISI)
Training based Channel estimation
After symbol timing offset and frame synchronization, the disc rete time received signal
• in frequency flat-fading channel is given by
𝑦 𝑛 = √(𝐸 )h𝑠 𝑛 + 𝑣 𝑛 h = 𝛼𝑒𝑗𝜙
• in frequency-selective fading environments is given by 𝐿
𝑦𝑛 =√(𝐸)h𝑙𝑠𝑛−𝑙 +𝑣[𝑛] 𝑥
Frequency flat channel estimation
Assume transmission of 𝑁 training symbols so that 𝑠 𝑛 = 𝑡[𝑛] 𝑡
and 𝑦[𝑛] is written as
𝑦𝑛= 𝐸𝛼𝑒𝑗𝜙𝑡𝑛+𝑣𝑛
The nth received signal during training substitute
𝑦[𝑛] = 𝑎𝑡[𝑛] + 𝑣[𝑚]
𝑎= 𝐸𝛼𝑒𝑗𝜙 𝑥
The objective is to estimate the unknown scalar channel ‘a’
Objective function
= (𝑦[𝑛]−𝑎𝑡[𝑛])∗(𝑦[𝑛]−𝑎𝑡[𝑛]) 𝑛=0
𝐽(a) = 𝑦[𝑛] − 𝑎𝑡[𝑛]
Optimization
Taking Partial derivative of the cost function w.r.t a∗, we have, 𝜕 𝜕 𝑁𝑡−1
𝜕a∗𝐽(a)=𝜕a∗ (𝑦[𝑛]−𝑎𝑡[𝑛])∗(𝑦 𝑛 −𝑎𝑡 𝑛 )
𝑁 −1 𝑛=0 𝜕𝑡
= 𝜕a∗ (𝑦∗[𝑛]𝑦[𝑛] − 𝑎∗𝑡∗[𝑛] 𝑦[𝑛] − 𝑎𝑡[𝑛]𝑦∗[𝑛] + 𝑎∗𝑎𝑡∗[𝑛]𝑡[𝑛]
𝑛=0 𝑁 −1 𝜕𝑡
𝜕a∗ 𝐽(a) = 𝑎𝑡∗ 𝑛 𝑡 𝑛 − 𝑡∗[𝑛]𝑦[𝑛] 𝑛=0
Sliding Correlator
The optimal Least squares estimate for the channel is obtained
𝑎𝑡∗ 𝑛 𝑡 𝑛 − 𝑡∗[𝑛]𝑦[𝑛] = 0
σ𝑁𝑡−1 𝑡∗[𝑛]𝑦[𝑛] h = 𝑛=0
𝐿𝑆 σ𝑁𝑡−1 𝑡∗ 𝑛 𝑡 𝑛 𝑛=0
h𝐿𝑆= (𝒕∗𝒕)−1(𝒕∗𝒚)
performs both frame sync and channel estimation
Frequency selective channel estimation
Least squares based channel estimation
• Signal model
Channel estimation
• LS solution is
Correlator – can jointly perform frame synchronization and channel estimation
Corrects for scaling
Blind frequency offset estimation for 4-QAM
Exploit symmetry in the 4-QAM constellation to develop a blind frequency offset estimator, Normalized 4-QAM constellation symbols are points on the unit circle. Assuming a static channel h
𝑗𝜋 Taking the fourth power that 𝑠𝟒 𝑛 = 𝑟𝑒 4
𝑦4𝑛=𝑒𝑗2𝜋𝜖4𝑛 h4⊗𝑠4𝑛 +𝑣4𝑛 𝑦4𝑛=𝑒𝑗2𝜋𝜖4𝑛 h4⊗−1+𝑣𝑛
=−𝑒𝑗2𝜋𝜖4𝑛 h4 +𝑣𝑛
Calculate the phase from
∠(𝑦4 𝑛+1 𝑦∗4 𝑛 )=∠𝑒 𝑗2𝜋𝜖4 𝑛+1 𝑒 −𝑗2𝜋𝜖4𝑛 =8𝜋𝜖+𝑣[𝑛]
𝜖Ƹ= 1 σ𝑁 ∠(𝑦4 𝑛+1 𝑦∗4 𝑛 8 𝜋(𝑁−1) 𝑛=1
Minimum Mean squared error (MMSE) Channel estimation
yn= htn+vn
Let g be an optimal MMSE chaanel estimate of h
The MMSE estimate of t = t[0] t[L +1] ... ˆ
t[n] = g* y[n] = ht[n]+ v[n] ˆ
e[n]= g*y[n]−t[n]
Taking the partial derivative w.r.t g* e[n] 2
g*y[n]−t[n] y[n]=0
g =(y*y]) t*y]
−1 gMMSE = (CYY ) CYt
Compare LS and MMSE channel Estimators
LS based channel estimate
h𝐿𝑆= (𝒕∗𝒕)−1(𝒕∗𝒚)
MMSE based channel estimate
y = ht + v gMMSE = (CYY )−1 CYt
C =y*y yy
C =y*t yt
= h*h+ n h*
and t*t=2
v*v=2 MMSE2 n
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