Post on 15-Oct-2021
transcript
Wireless Communications From 5G and WiFi 6 to Low Power IoT
Lecture 4: OFDMHaitham Hassanieh
Yesterday’s Lecture Was
A. Too Difficult: I understood nothing.
B. Difficult: I understood something but not everything.
C. Moderate: I understood most stuff but I have questions
D. Easy: I understood everything really well.
Multipath in the Wireless Channel is problematic since it creates:
A. Inter-Symbol-Interference
B. Pathloss
C. Frequency Selective Fading
D. Additive White Gaussian Noise
ISI is considered negligible if the delayed symbols arriving longer paths interfere with less than < 1% of the symbol length.
If the longest path in the channel delays the symbol by 10ns, what is the maximum bandwidth for which we can ignore ISI?
A. 1 MHz
B. 10 MHz
C. 100 MHz
D. You can never ignore ISI
Previous Lecture:ü Pulse Shaping
ü Matched Filter
ü Multipath Channel
ü Channel Estimation & Correction
ü Narrowband vs. Wideband Channels
ü Channel Equalization
q Multi-Carrier Modulation
q Orthogonal Frequency Division Multiplexing (OFDM)
q OFDM Time Synchronization
q OFDM Frequency Synchronization
q OFDM Channel Estimation & Correction
q OFDM Phase Tracking
This Lecture:
Wireless Communication System
Symbols-to-Bits Mapper
Bits
Synchronization Demodulation(Decoding)
MatchedFilter
TX RX
10110011001 10110111001
Bits-to-SymbolsMapper
LPF BPF
PA
PLL
Mixer
Pulse Shaping
DACModulation (Encoding)
LPFBPFLNA
PLL
Mixer
ADC ChannelEqualization
ℎ!" 𝑡
Training + Data Bits
Wireless Communication System
Symbols-to-Bits Mapper
Bits
Demodulation(Decoding)
MatchedFilter
TX RX
10110011001 10110111001
Bits-to-SymbolsMapper
LPF BPF
PA
PLL
Mixer
Pulse Shaping
DACModulation (Encoding)
LPFBPFLNA
PLL
Mixer
ADC ChannelEqualization
ℎ!" 𝑡
Training + Data Bits
Δ𝑓!Frequency
SynchronizationTime Sync.
Wireless Communication System
Symbols-to-Bits Mapper
Bits
Demodulation(Decoding)
MatchedFilter
Bits-to-SymbolsMapper
LPF BPF
PA
PLL
Mixer
Pulse Shaping
DACModulation (Encoding)
LPFBPFLNA
PLL
Mixer
ADC ChannelEqualization
ℎ!" 𝑡
Training + Data Bits
Δ𝑓!Frequency
SynchronizationTime Sync.
TX
RX
Single Carrier ModulationSymbols modulated on a single carrier frequency
𝑠 𝑛 cos 2𝜋𝑓!𝑡
Single Carrier ModulationSymbols modulated on a single carrier frequency
• Low Spectral Efficiency: sinc & raised cosine leakage
• ISI: Inter-Symbol-Interference limits performance
𝑓
−𝐵 𝐵
Thus, for narrow band, convolving with the wireless channel reduces to multiplying by a singlecomplex number h and we can now write the received signal y(t) as:
y(t) = hx(t) + n(t).
• Wide Band Channel: For wide band we can approximate the wireless channel h(t) by amulti-tap channel i.e. multiple delayed impulses as shown in Figure 5. For a k tap channelthe received signal y(t) can be written as:
y(t) =i=k∑
i=0
h(i)s(t− iτ)
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60 70
|H|2
Tap Index
Figure 5: Time Domain Wide Band Channel h(t) (≈11 taps)
• Frequency Selective Fading: Convolution with h(t) in the time domain results multipli-cation with H(f) in the frequency domain. For narrow band, h(t) is an impulse and H(f) isflat. For wide band, H(f) results in different attenuation for different frequencies as shownin Figure 6. The figure also shows that for narrow bands the channel can be approximated asflat.
y(t) = h(t) ∗ s(t) + n(t) ⇔ Y (f) = H(f)S(f) +N
−80 −60 −40 −20 0 20 40 60 80Frequency in MHz
Figure 6: Frequency Selective Fading for 100 MHz channel
• Inter-Symbol-Interference: Multi-path results in inter-symbol-interference i.e. delayedsymbols interfere with the symbol being decoding. The effect is sever and results in decodingerrors for wide band since the symbol length is short and of the order of the delayed taps. Thenext lecture will discuss how we deal with this problem using OFDM (Orthogonal FrequencyDivision Multiplexing).
Thus, for narrow band, convolving with the wireless channel reduces to multiplying by a singlecomplex number h and we can now write the received signal y(t) as:
y(t) = hx(t) + n(t).
• Wide Band Channel: For wide band we can approximate the wireless channel h(t) by amulti-tap channel i.e. multiple delayed impulses as shown in Figure 5. For a k tap channelthe received signal y(t) can be written as:
y(t) =i=k∑
i=0
h(i)s(t− iτ)
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60 70
|H|2
Tap Index
Figure 5: Time Domain Wide Band Channel h(t) (≈11 taps)
• Frequency Selective Fading: Convolution with h(t) in the time domain results multipli-cation with H(f) in the frequency domain. For narrow band, h(t) is an impulse and H(f) isflat. For wide band, H(f) results in different attenuation for different frequencies as shownin Figure 6. The figure also shows that for narrow bands the channel can be approximated asflat.
y(t) = h(t) ∗ s(t) + n(t) ⇔ Y (f) = H(f)S(f) +N
−80 −60 −40 −20 0 20 40 60 80Frequency in MHz
Figure 6: Frequency Selective Fading for 100 MHz channel
• Inter-Symbol-Interference: Multi-path results in inter-symbol-interference i.e. delayedsymbols interfere with the symbol being decoding. The effect is sever and results in decodingerrors for wide band since the symbol length is short and of the order of the delayed taps. Thenext lecture will discuss how we deal with this problem using OFDM (Orthogonal FrequencyDivision Multiplexing).
Multi-Carrier Modulation
• Divide spectrum into many narrow bands
• Transmit symbols on different carriers in narrow bands
Symbols modulated on multiple Sub-carrier frequencies
Thus, for narrow band, convolving with the wireless channel reduces to multiplying by a singlecomplex number h and we can now write the received signal y(t) as:
y(t) = hx(t) + n(t).
• Wide Band Channel: For wide band we can approximate the wireless channel h(t) by amulti-tap channel i.e. multiple delayed impulses as shown in Figure 5. For a k tap channelthe received signal y(t) can be written as:
y(t) =i=k∑
i=0
h(i)s(t− iτ)
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60 70
|H|2
Tap Index
Figure 5: Time Domain Wide Band Channel h(t) (≈11 taps)
• Frequency Selective Fading: Convolution with h(t) in the time domain results multipli-cation with H(f) in the frequency domain. For narrow band, h(t) is an impulse and H(f) isflat. For wide band, H(f) results in different attenuation for different frequencies as shownin Figure 6. The figure also shows that for narrow bands the channel can be approximated asflat.
y(t) = h(t) ∗ s(t) + n(t) ⇔ Y (f) = H(f)S(f) +N
−80 −60 −40 −20 0 20 40 60 80Frequency in MHz
Figure 6: Frequency Selective Fading for 100 MHz channel
• Inter-Symbol-Interference: Multi-path results in inter-symbol-interference i.e. delayedsymbols interfere with the symbol being decoding. The effect is sever and results in decodingerrors for wide band since the symbol length is short and of the order of the delayed taps. Thenext lecture will discuss how we deal with this problem using OFDM (Orthogonal FrequencyDivision Multiplexing).
• Channel is Flat à No need to worry about ISI
𝑥 𝑡 =$#
𝑠# 𝑛 e$%&'(!)
Multi-Carrier Modulation
• Divide spectrum into many narrow bands
Thus, for narrow band, convolving with the wireless channel reduces to multiplying by a singlecomplex number h and we can now write the received signal y(t) as:
y(t) = hx(t) + n(t).
• Wide Band Channel: For wide band we can approximate the wireless channel h(t) by amulti-tap channel i.e. multiple delayed impulses as shown in Figure 5. For a k tap channelthe received signal y(t) can be written as:
y(t) =i=k∑
i=0
h(i)s(t− iτ)
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60 70
|H|2
Tap Index
Figure 5: Time Domain Wide Band Channel h(t) (≈11 taps)
• Frequency Selective Fading: Convolution with h(t) in the time domain results multipli-cation with H(f) in the frequency domain. For narrow band, h(t) is an impulse and H(f) isflat. For wide band, H(f) results in different attenuation for different frequencies as shownin Figure 6. The figure also shows that for narrow bands the channel can be approximated asflat.
y(t) = h(t) ∗ s(t) + n(t) ⇔ Y (f) = H(f)S(f) +N
−80 −60 −40 −20 0 20 40 60 80Frequency in MHz
Figure 6: Frequency Selective Fading for 100 MHz channel
• Inter-Symbol-Interference: Multi-path results in inter-symbol-interference i.e. delayedsymbols interfere with the symbol being decoding. The effect is sever and results in decodingerrors for wide band since the symbol length is short and of the order of the delayed taps. Thenext lecture will discuss how we deal with this problem using OFDM (Orthogonal FrequencyDivision Multiplexing).
Symbols modulated on multiple Sub-carrier frequencies
• Transmit symbols on different carriers in narrow bands
• Channel is Flat à No need to worry about ISI
𝑥 𝑡 =$#
𝑠# 𝑛 e$%&'(!)
𝑦 𝑡 =$#
ℎ#𝑠# 𝑛 e$%&'(!)
Multi-Carrier Modulation
• Divide spectrum into many narrow bands
Thus, for narrow band, convolving with the wireless channel reduces to multiplying by a singlecomplex number h and we can now write the received signal y(t) as:
y(t) = hx(t) + n(t).
• Wide Band Channel: For wide band we can approximate the wireless channel h(t) by amulti-tap channel i.e. multiple delayed impulses as shown in Figure 5. For a k tap channelthe received signal y(t) can be written as:
y(t) =i=k∑
i=0
h(i)s(t− iτ)
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60 70
|H|2
Tap Index
Figure 5: Time Domain Wide Band Channel h(t) (≈11 taps)
• Frequency Selective Fading: Convolution with h(t) in the time domain results multipli-cation with H(f) in the frequency domain. For narrow band, h(t) is an impulse and H(f) isflat. For wide band, H(f) results in different attenuation for different frequencies as shownin Figure 6. The figure also shows that for narrow bands the channel can be approximated asflat.
y(t) = h(t) ∗ s(t) + n(t) ⇔ Y (f) = H(f)S(f) +N
−80 −60 −40 −20 0 20 40 60 80Frequency in MHz
Figure 6: Frequency Selective Fading for 100 MHz channel
• Inter-Symbol-Interference: Multi-path results in inter-symbol-interference i.e. delayedsymbols interfere with the symbol being decoding. The effect is sever and results in decodingerrors for wide band since the symbol length is short and of the order of the delayed taps. Thenext lecture will discuss how we deal with this problem using OFDM (Orthogonal FrequencyDivision Multiplexing).
Symbols modulated on multiple Sub-carrier frequencies
• Transmit symbols on different carriers in narrow bands
𝑥 𝑡 =$#
𝑠# 𝑛 e$%&'(!)
• Channel is Flat à No need to worry about ISI
𝑦 𝑡 =$#
ℎ#𝑠# 𝑛 e$%&'(!)
Not That Simple!
Multi-Carrier ModulationSymbols modulated on multiple Sub-carrier frequencies
• Divide spectrum into many narrow bands
• Significant Leakage between adjacent subcarriers
• Need Guard Bands à Very inefficient!
𝑓 𝑓
Guard Bands
Solution: Make the Sub-Carriers Orthogonal
Multi-Carrier ModulationSymbols modulated on multiple Sub-carrier frequencies
Make the Sub-Carriers Orthogonal
OFDM: Orthogonal Frequency Division Multiplexing
• Subcarriers are orthogonal: At the sub-carrier frequency, the sampled value has zero leakage from other subcarriers.
• Subcarrier separation can be very small, for N subcarriers and bandwidth B:
Δ𝑓 =𝐵𝑁
OFDM: Orthogonal Frequency Division Multiplexing
• Subcarriers are orthogonal: At the sub-carrier frequency, the sampled value has zero leakage from other subcarriers.
• Subcarrier separation can be very small, for N subcarriers and bandwidth B:
Δ𝑓 =𝐵𝑁
How to Achieve This?
OFDM: Orthogonal Frequency Division Multiplexing
N-Point DFT:
Use DFT: Discrete Fourier Transform
𝑋 𝑓# =1𝑁$)*+
,$-
𝑥 𝑡 𝑒$%&'(!),
𝑥(𝑡) = $(!*+
,$-
𝑋 𝑓# 𝑒%&'(!),N-Point IDFT:
Send symbols in Frequency Domain𝑋 𝑓" = 𝑠 𝑛 → Compute and transmit 𝑥 𝑡 using IDFT
OFDM: Orthogonal Frequency Division Multiplexing
Send symbols in Frequency Domain𝑋 𝑓" = 𝑠 𝑛 → Compute and transmit 𝑥 𝑡 using IDFT
• 𝑁subcarrier à IDFT of length 𝑁
• Symbols 𝑠 𝑛 can come from any modulation: BPSK, QPSK, QAM…
• 𝑥 𝑡 is complex à need 𝐼 & 𝑄à No point using PAM or ASK …
• OFDM Symbol: 𝑁 samples of 𝑥 𝑡 generated from the same modulated symbols using IDFT.
• OFDM Symbol Time: 𝑇 = 𝑁/𝐵 where 𝐵 is the bandwidth.
• OFDM Frequency Bin Width: Δ𝑓 = 1/𝑇 = 𝐵/𝑁
Receiver
Transmitter
ModulationBits
Demodulation Bits
OFDM: Orthogonal Frequency Division Multiplexing
Para
llel t
o Se
rial
Seria
l to
Para
llel
IFFT
Para
llel t
o Se
rial
Seria
l to
Para
llel
FFT
LPFBPF
PLL
Mixer
LPFBPF Mixer
90"LNA
ADC
ADC
𝐼
𝑗𝑄+
LPF BPFMixerDAC
LPF BPF
PLL
Mixer
90"PA
DAC
ℜ𝔢{ }
ℑ𝔪{ }
+
OFDM Symbol in Frequency Domain
Thus, for narrow band, convolving with the wireless channel reduces to multiplying by a singlecomplex number h and we can now write the received signal y(t) as:
y(t) = hx(t) + n(t).
• Wide Band Channel: For wide band we can approximate the wireless channel h(t) by amulti-tap channel i.e. multiple delayed impulses as shown in Figure 5. For a k tap channelthe received signal y(t) can be written as:
y(t) =i=k∑
i=0
h(i)s(t− iτ)
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60 70
|H|2
Tap Index
Figure 5: Time Domain Wide Band Channel h(t) (≈11 taps)
• Frequency Selective Fading: Convolution with h(t) in the time domain results multipli-cation with H(f) in the frequency domain. For narrow band, h(t) is an impulse and H(f) isflat. For wide band, H(f) results in different attenuation for different frequencies as shownin Figure 6. The figure also shows that for narrow bands the channel can be approximated asflat.
y(t) = h(t) ∗ s(t) + n(t) ⇔ Y (f) = H(f)S(f) +N
−80 −60 −40 −20 0 20 40 60 80Frequency in MHz
Figure 6: Frequency Selective Fading for 100 MHz channel
• Inter-Symbol-Interference: Multi-path results in inter-symbol-interference i.e. delayedsymbols interfere with the symbol being decoding. The effect is sever and results in decodingerrors for wide band since the symbol length is short and of the order of the delayed taps. Thenext lecture will discuss how we deal with this problem using OFDM (Orthogonal FrequencyDivision Multiplexing).
0−𝑁2
𝑁2− 1
• FFT can be represented 0 to 𝑁 − 1 or 𝑁/2 to 𝑁/2 − 1.
• OFDM Symbol created in digital baseband à 0 bin corresponds to DC
𝑋 0 =1𝑁$)*+
,$-
𝑥 𝑡 𝑒$%&'+), =
1𝑁$)*+
,$-
𝑥 𝑡 = 𝐷𝐶
OFDM Symbol in Frequency Domain
Thus, for narrow band, convolving with the wireless channel reduces to multiplying by a singlecomplex number h and we can now write the received signal y(t) as:
y(t) = hx(t) + n(t).
• Wide Band Channel: For wide band we can approximate the wireless channel h(t) by amulti-tap channel i.e. multiple delayed impulses as shown in Figure 5. For a k tap channelthe received signal y(t) can be written as:
y(t) =i=k∑
i=0
h(i)s(t− iτ)
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60 70
|H|2
Tap Index
Figure 5: Time Domain Wide Band Channel h(t) (≈11 taps)
• Frequency Selective Fading: Convolution with h(t) in the time domain results multipli-cation with H(f) in the frequency domain. For narrow band, h(t) is an impulse and H(f) isflat. For wide band, H(f) results in different attenuation for different frequencies as shownin Figure 6. The figure also shows that for narrow bands the channel can be approximated asflat.
y(t) = h(t) ∗ s(t) + n(t) ⇔ Y (f) = H(f)S(f) +N
−80 −60 −40 −20 0 20 40 60 80Frequency in MHz
Figure 6: Frequency Selective Fading for 100 MHz channel
• Inter-Symbol-Interference: Multi-path results in inter-symbol-interference i.e. delayedsymbols interfere with the symbol being decoding. The effect is sever and results in decodingerrors for wide band since the symbol length is short and of the order of the delayed taps. Thenext lecture will discuss how we deal with this problem using OFDM (Orthogonal FrequencyDivision Multiplexing).
0−𝑁2
𝑁2− 1
• FFT can be represented 0 to 𝑁 − 1 or 𝑁/2 to 𝑁/2 − 1.
• OFDM Symbol created in digital baseband à 0 bin corresponds to DC
• DC of the circuits corrupts bits sent on the 0 bin à Do not use 0 bin
OFDM Symbol in Frequency Domain
Thus, for narrow band, convolving with the wireless channel reduces to multiplying by a singlecomplex number h and we can now write the received signal y(t) as:
y(t) = hx(t) + n(t).
• Wide Band Channel: For wide band we can approximate the wireless channel h(t) by amulti-tap channel i.e. multiple delayed impulses as shown in Figure 5. For a k tap channelthe received signal y(t) can be written as:
y(t) =i=k∑
i=0
h(i)s(t− iτ)
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60 70
|H|2
Tap Index
Figure 5: Time Domain Wide Band Channel h(t) (≈11 taps)
• Frequency Selective Fading: Convolution with h(t) in the time domain results multipli-cation with H(f) in the frequency domain. For narrow band, h(t) is an impulse and H(f) isflat. For wide band, H(f) results in different attenuation for different frequencies as shownin Figure 6. The figure also shows that for narrow bands the channel can be approximated asflat.
y(t) = h(t) ∗ s(t) + n(t) ⇔ Y (f) = H(f)S(f) +N
−80 −60 −40 −20 0 20 40 60 80Frequency in MHz
Figure 6: Frequency Selective Fading for 100 MHz channel
• Inter-Symbol-Interference: Multi-path results in inter-symbol-interference i.e. delayedsymbols interfere with the symbol being decoding. The effect is sever and results in decodingerrors for wide band since the symbol length is short and of the order of the delayed taps. Thenext lecture will discuss how we deal with this problem using OFDM (Orthogonal FrequencyDivision Multiplexing).
−𝑁2
𝑁2− 1
• FFT can be represented 0 to 𝑁 − 1 or 𝑁/2 to 𝑁/2 − 1.
• OFDM Symbol created in digital baseband à 0 bin corresponds to DC
• DC of the circuits corrupts bits sent on the 0 bin à Do not use 0 bin
0
DC Bin
OFDM Symbol in Frequency Domain
Thus, for narrow band, convolving with the wireless channel reduces to multiplying by a singlecomplex number h and we can now write the received signal y(t) as:
y(t) = hx(t) + n(t).
• Wide Band Channel: For wide band we can approximate the wireless channel h(t) by amulti-tap channel i.e. multiple delayed impulses as shown in Figure 5. For a k tap channelthe received signal y(t) can be written as:
y(t) =i=k∑
i=0
h(i)s(t− iτ)
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60 70
|H|2
Tap Index
Figure 5: Time Domain Wide Band Channel h(t) (≈11 taps)
• Frequency Selective Fading: Convolution with h(t) in the time domain results multipli-cation with H(f) in the frequency domain. For narrow band, h(t) is an impulse and H(f) isflat. For wide band, H(f) results in different attenuation for different frequencies as shownin Figure 6. The figure also shows that for narrow bands the channel can be approximated asflat.
y(t) = h(t) ∗ s(t) + n(t) ⇔ Y (f) = H(f)S(f) +N
−80 −60 −40 −20 0 20 40 60 80Frequency in MHz
Figure 6: Frequency Selective Fading for 100 MHz channel
• Inter-Symbol-Interference: Multi-path results in inter-symbol-interference i.e. delayedsymbols interfere with the symbol being decoding. The effect is sever and results in decodingerrors for wide band since the symbol length is short and of the order of the delayed taps. Thenext lecture will discuss how we deal with this problem using OFDM (Orthogonal FrequencyDivision Multiplexing).
0−𝑁2
𝑁2− 1
• Subcarriers orthogonal to each other but not to near by channels.
• Need Guard Bins at sides of the channel à Transmit nothing there
DC Bin
OFDM Symbol in Frequency Domain
Thus, for narrow band, convolving with the wireless channel reduces to multiplying by a singlecomplex number h and we can now write the received signal y(t) as:
y(t) = hx(t) + n(t).
• Wide Band Channel: For wide band we can approximate the wireless channel h(t) by amulti-tap channel i.e. multiple delayed impulses as shown in Figure 5. For a k tap channelthe received signal y(t) can be written as:
y(t) =i=k∑
i=0
h(i)s(t− iτ)
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60 70
|H|2
Tap Index
Figure 5: Time Domain Wide Band Channel h(t) (≈11 taps)
• Frequency Selective Fading: Convolution with h(t) in the time domain results multipli-cation with H(f) in the frequency domain. For narrow band, h(t) is an impulse and H(f) isflat. For wide band, H(f) results in different attenuation for different frequencies as shownin Figure 6. The figure also shows that for narrow bands the channel can be approximated asflat.
y(t) = h(t) ∗ s(t) + n(t) ⇔ Y (f) = H(f)S(f) +N
−80 −60 −40 −20 0 20 40 60 80Frequency in MHz
Figure 6: Frequency Selective Fading for 100 MHz channel
• Inter-Symbol-Interference: Multi-path results in inter-symbol-interference i.e. delayedsymbols interfere with the symbol being decoding. The effect is sever and results in decodingerrors for wide band since the symbol length is short and of the order of the delayed taps. Thenext lecture will discuss how we deal with this problem using OFDM (Orthogonal FrequencyDivision Multiplexing).
0−𝑁2
𝑁2− 1
• Subcarriers orthogonal to each other but not to near by channels.
• Need Guard Bins at sides of the channel à Transmit nothing there
• Reduce Number of Guard band from 𝑁 to 2 à Very Spectrally Efficient
Guard BinsGuard Bins DC Bin
ModulationBits
Demodulation Bits
OFDM: Orthogonal Frequency Division Multiplexing
Para
llel t
o Se
rial
Seria
l to
Para
llel
IFFT
Para
llel t
o Se
rial
Seria
l to
Para
llel
FFT
LPFBPF
PLL
Mixer
LPFBPF Mixer
90"LNA
ADC
ADC
𝐼
𝑗𝑄+
LPF BPFMixerDAC
LPF BPF
PLL
Mixer
90"PA
DAC
ℜ𝔢{ }
ℑ𝔪{ }
+TX
RX
Transmit Symbols in Frequency Domain On Orthogonal Subcarriers
OFDM Symbol
0−𝑁2
𝑁2 − 1
Guard Bins
Guard Bins
DC
… ,+1,−1,+1,−1,+1,−1,−1,−1,+1,+1,−1,+1,+1,−1,−1,…
1 0 1 0 1 0 0 0 1 1 0 1 1 0 0Bits:
IFFT
Symbol in Time
OFDM Symbol
Symbol in Time Symbol in Time Symbol in Time Symbol in Time ⋯⋯FFT FFT FFT FFT
… ,+1,−1,+1,… … ,+1,+1,+1,… … ,−1,−1,+1,… … ,+1,−1,−1,…
…101… …111… …001… …100…
Not That Simple
OFDM Symbol
S1 S2 S3 S4 ⋯⋯
FFT
… ,+1,−1,+1,…
…101…
FFT Window
OFDM Symbol
S1 S2 S3 S4 ⋯⋯
FFT FFT
… ,+1,−1,+1,… … ,+1,+1,+1,…
…101… …111…
FFT Window
OFDM Symbol
S1 S2 S3 S4 ⋯⋯
FFT FFT FFT
… ,+1,−1,+1,… … ,+1,+1,+1,… … ,−1,−1,+1,…
…101… …111… …001…
FFT Window
OFDM Symbol
S1 S2 S3 S4 ⋯⋯
FFT FFT FFT FFT
… ,+1,−1,+1,… … ,+1,+1,+1,… … ,−1,−1,+1,… … ,+1,−1,−1,…
…101… …111… …001… …100…
FFT Window
Assumes FFT window is perfectly aligned with symbol boundaries
OFDM Symbol
S1 S2 S3 S4 ⋯⋯
FFT
… ,+0.5 + 1i, −0.7 + 0.3i, …
FFT Window
FFT window is misaligned with symbol
Cannot decode!
Subcarriers are no longer orthogonal.
OFDM Cyclic Prefix
FFT Window
• DFT (FFT) assumes time samples are periodic of period 𝑁
𝑥[𝑡] → 𝑋[𝑓]
𝑥 𝑡 − 𝜏 mod 𝑁 → 𝑋 𝑓 𝑒0123456
• Circular Shift before taking FFT:
S1 S2 S3 S4 ⋯⋯
OFDM Cyclic Prefix
S1 S2 S3FFT Window
S1 S2 S3
• DFT (FFT) assumes time samples are periodic of period 𝑁
𝑥[𝑡] → 𝑋[𝑓]
𝑥 𝑡 − 𝜏 mod 𝑁 → 𝑋 𝑓 𝑒0123456
• Circular Shift before taking FFT:
OFDM Cyclic Prefix
S1 S2 S3FFT Window
S1 S2 S3
• Even if FFT window is misaligned, CP ensures that all samples come from the same symbol à Orthogonality is preserved!
• Cyclic Prefix can be created by:o Take first few samples and append them to end of symbol.o Take last few samples and prefix them to beginning of symbol.
• Simple Phase Shift à Can be corrected by lumping with channel 𝐻[𝑓]
OFDM Cyclic Prefix
S1 S2 S3FFT Window
S1 S2 S3
Cyclic Prefix:
• Preserves orthogonality by allowing some misalignment in FFT Window
• Deals with Inter-Symbol-Interference
ISI ISI ISI
OFDM Cyclic Prefix
S1 S2 S3FFT Window
S1 S2 S3
Cyclic Prefix:
• Preserves orthogonality by allowing some misalignment in FFT Window
• Deals with Inter-Symbol-Interference
ISI ISI ISI
NO ISI inFFT Window
OFDM Cyclic Prefix
S1 S2 S3FFT Window
S1 S2 S3
Cyclic Prefix:
• Preserves orthogonality by allowing some misalignment in FFT Window
• Deals with Inter-Symbol-Interference
ISI ISI ISI
NO ISI inFFT Window
FFT Window
NO ISI inFFT Window
FFT Window
NO ISI inFFT Window
• Overhead: Send 𝐶𝑃 + 𝑁 samples for every 𝑁 samples
OFDM Cyclic PrefixCyclic Prefix:
• Preserves orthogonality by allowing some misalignment in FFT Window
• Deals with Inter-Symbol-Interference
Overhead =𝐶𝑃
𝐶𝑃 + 𝑁
e. g.WiFi 802.11n:𝑁 = 64, CP = 16 → Overhead = 20%
e. g. LTE:𝑁 = 1024, CP = 72 → Overhead = 6.5%
Progress so far?
A. I understand and I can follow
B. I understand most stuff but not everything
C. I can follow but I do not understand everything
D. I cannot follow and understood nothing about OFDM
OFDM Cyclic Prefix
S1 S2 S3FFT Window
S1 S2 S3
• Cyclic prefix is a not a bullet proof solution.
• Can still end up misaligned!
• Need a way to ensure we detect the beginning of the packet correctly.
• If we do, CP will ensure that even if we are not accurate, we can still decode.
OFDM Packet Detection
• Detect Beginning of packet to make sure we are within the CP
• Send Training Sequence: Preamble Symbols
• Preamble Symbols: Known Symbol Repeated at the beginning of packet
S1 S2CP1 CP2Preamble Preamble Preamble ⋯ ⋯
• No need for CP with preamble symbols
OFDM Packet Detection: Sliding Window
S1 S2CP1Preamble Preamble Preamble ⋯A B
• Two windows of 𝐿 (2𝑁) samples each.
• Compute:𝑃7𝑃8
=∑9:;<=;<2= 𝑥[𝑘] 2
∑9:;;<= 𝑥[𝑘] 2
𝑃!𝑃"
1
OFDM Packet Detection: Sliding Window
S1 S2CP1Preamble Preamble Preamble ⋯A B
• Two windows of 𝐿 (2𝑁) samples each.
• Compute:𝑃7𝑃8
=∑9:;<=;<2= 𝑥[𝑘] 2
∑9:;;<= 𝑥[𝑘] 2
𝑃!𝑃"
1
OFDM Packet Detection: Sliding Window
S1 S2CP1Preamble Preamble Preamble ⋯A B
• Two windows of 𝐿 samples each.
• Compute:𝑃7𝑃8
=∑9:;<=;<2= 𝑥[𝑘] 2
∑9:;;<= 𝑥[𝑘] 2
𝑃!𝑃"
1
OFDM Packet Detection: Sliding Window
S1 S2CP1Preamble Preamble Preamble ⋯A B
• Two windows of 𝐿 samples each.
• Compute:𝑃7𝑃8
=∑9:;<=;<2= 𝑥[𝑘] 2
∑9:;;<= 𝑥[𝑘] 2
𝑃!𝑃"
1
OFDM Packet Detection: Sliding Window
S1 S2CP1Preamble Preamble Preamble ⋯A B
• Two windows of 𝐿 samples each.
• Compute:𝑃7𝑃8
=∑9:;<=;<2= 𝑥[𝑘] 2
∑9:;;<= 𝑥[𝑘] 2
𝑃!𝑃"
1
OFDM Packet Detection: Sliding Window
S1 S2CP1Preamble Preamble Preamble ⋯A B
• Two windows of 𝐿 samples each.
• Compute:𝑃7𝑃8
=∑9:;<=;<2= 𝑥[𝑘] 2
∑9:;;<= 𝑥[𝑘] 2
𝑃!𝑃"
1
Packet Start +L
Previous Lecture:ü Pulse Shaping
ü Matched Filter
ü Multipath Channel
ü Channel Estimation & Correction
ü Narrowband vs. Wideband Channels
ü Channel Equalization
ü Multi-Carrier Modulation
ü Orthogonal Frequency Division Multiplexing (OFDM)
ü OFDM Time Synchronization
q OFDM Frequency Synchronization
q OFDM Channel Estimation & Correction
q OFDM Phase Tracking
This Lecture:
ModulationBits
Demodulation Bits
OFDM: Orthogonal Frequency Division Multiplexing
Para
llel t
o Se
rial
Seria
l to
Para
llel
IFFT
Para
llel t
o Se
rial
Seria
l to
Para
llel
FFT
LPFBPF
PLL
Mixer
LPFBPF Mixer
90"LNA
ADC
ADC
𝐼
𝑗𝑄+
LPF BPFMixerDAC
LPF BPF
PLL
Mixer
90"PA
DAC
ℜ𝔢{ }
ℑ𝔪{ }
+TX
RX
Transmit Symbols in Frequency Domain On Orthogonal Subcarriers
So far, we assumed carriers generated by LOs are
perfectly synchronized!
Carrier Frequency Offset
TX RX
101101011 101101011
𝑥 𝑡 ℎ 𝑡 ∗ 𝑥 𝑡 𝑒#$%&'#(𝑥 𝑡 ×𝑒#$%&'#( ℎ 𝑡 ∗ 𝑥 𝑡 𝑒#$%&'#(×𝑒$%&'#(
ℎ 𝑡 ∗ 𝑥 𝑡
𝑦 𝑡 = ℎ 𝑡 ∗ 𝑥 𝑡 + 𝑣 𝑡
Assumes TX & RX perfectly synched
Carrier Frequency Offset
TX RX
101101011 101101011
𝑥 𝑡 ℎ 𝑡 ∗ 𝑥 𝑡 𝑒#$%&'#(𝑥 𝑡 ×𝑒#$%&'#( ℎ 𝑡 ∗ 𝑥 𝑡 𝑒#$%&'#(×𝑒$%&'#$(
ℎ 𝑡 ∗ 𝑥 𝑡 𝑒#$%&)'#(
𝑦 𝑡 = ℎ 𝑡 ∗ 𝑥 𝑡 𝑒#$%&)'#( + 𝑣 𝑡
TX & RX are not synched
Phase changes with time!
CFO: Δ𝑓0 = 𝑓0 − 𝑓01
Carrier Frequency Offset
+1−1 𝐼
𝑄
Consider BPSK Modulation.0 → −11 → +1
𝑥 𝑡 ℎ 𝑥 𝑡 − 𝜏 𝑒$%&'2(+) + 𝑣 𝑡
+1−1 𝐼
𝑄
Carrier Frequency Offset
+1−1 𝐼
𝑄
Consider BPSK Modulation.0 → −11 → +1
𝑥 𝑡 ℎ 𝑥 𝑡 − 𝜏 𝑒$%&'2(+) + 𝑣 𝑡
+1−1 𝐼
𝑄
Impossible to Decode!
Carrier Frequency Offset
𝐼
𝑄Consider 16 QAM Modulation
Need to estimate and correct CFO to decode!
OFDM CFO Estimation & Correction
• Use Preamble to estimate CFO
𝑦H 𝑡 = 𝑥 𝑡 𝑒0123I4!;
𝑦2 𝑡 = 𝑥 𝑡 𝑒0123I4! ;<6J"
S1 S2CP1 CP2Preamble Preamble Preamble ⋯ ⋯
𝑦H[𝑛] = 𝑥[𝑛]𝑒0123I4!KJ"
𝑦2[𝑛] = 𝑥[𝑛]𝑒0123I4! KJ"<6J"
OFDM CFO Estimation & Correction
• Use Preamble to estimate CFO
• Compute: 𝐴 =$)*-
,
𝑦- 𝑛 𝑦&∗[𝑛] =$)*-
,
𝑥 𝑛 𝑥∗[𝑛]𝑒%&'2(+,4,
= 𝑒%&'2(+,4,$)*-
,
𝑥 𝑛 & Δ𝑓0 =∠𝐴
2𝜋𝑁𝑇5
S1 S2CP1 CP2Preamble Preamble Preamble ⋯ ⋯
𝑦H[𝑛] = 𝑥[𝑛]𝑒0123I4!KJ"
𝑦2[𝑛] = 𝑥[𝑛]𝑒0123I4! KJ"<6J"
OFDM CFO Estimation & Correction
• Use Preamble to estimate CFO
• Compute: 𝐴 =$)*-
,
𝑦- 𝑛 𝑦&∗[𝑛] Δ𝑓0 =∠𝐴
2𝜋𝑁𝑇5
• Correct CFO: 𝑦 𝑛 ×𝑒123I4!KJ"
S1 S2CP1 CP2Preamble Preamble Preamble ⋯ ⋯
We use the following equation to estimate CFO: Δ𝑓0 =∠7
&',4*.
Suppose 𝑓0 = 5 GHz, the bandwidth = 10 MHz and the clock precision is 20ppm. For what values of N will the above equation
estimate the CFO incorrectly?
A. N < 10
B. N < 20
C. N > 50
D. N < 50
• Equation give wrong result when the phase of A wraps around 2𝜋! We need: ∠𝐴 ≤ 𝜋
Δ𝑓0 ≤1
2𝑁𝑇5
• Δ𝑓0 = 5 GHz × 20/1000000 = 100 kHz
• 𝑇5 = 1/10MHz = 0.1𝜇𝑠
OFDM Channel Estimation
• Use Preamble to estimate the channel
𝑦 𝑡 = ℎ 𝑡 ∗ 𝑥 𝑡 ↔ 𝑌 𝑓 = 𝐻 𝑓 𝑋 𝑓
• Send 𝑋 𝑓 : −1,+1,−1,−1,−1,+1,…
• Receive: −𝐻(1), 𝐻(2), −𝐻(3), −𝐻(4), −𝐻(5), 𝐻(6), …
• Estimate: Z𝐻 𝑓 =𝑌 𝑓𝑋 𝑓
• Use two preambles to average noise: Z𝐻 𝑓 =𝑌H 𝑓 + 𝑌2 𝑓
2 𝑋 𝑓
S1 S2CP1 CP2Preamble Preamble Preamble ⋯ ⋯
Phase TrackingSo Far: Estimated and Corrected For Coarse Value of CFO
• Residual CFO:
𝑦 𝑡 = ℎ 𝑡 𝑥 𝑡 𝑒$%&'2(+) + 𝑣 𝑡
Δ𝑓0 = 𝑑𝑓0 + 𝛿𝑓0
Coarse CFO Residual CFO
We estimated and corrected for coarse CFO!
Even small residual can accumulate over time to create large phase: 𝑒$%&'8(+)
Need to track the phase
Phase Tracking• Residual CFO (Carrier Frequency Offset)
• Residual SFO (Sampling Frequency Offset)
Phase Tracking
𝑦[𝑛] = 𝑥[𝑛 + 𝑛𝛿𝑇L]𝑒0123M4!KJ"
= ]4:N
60H
𝑋 𝑓 𝑒1234(K<KMJ")
6 𝑒0123M4!KJ"
𝑌 𝑓 = 𝑋[𝑓]𝑒1234KMJ"
6 023M4!KJ"
• Residual CFO (Carrier Frequency Offset)
• Residual SFO (Sampling Frequency Offset)
When we sample the signal there is a residual sampling offset: 𝑛𝛿𝑇5
Phase Tracking
𝑌H 𝑓 = 𝑋H[𝑓]𝑒1234KMJ"6 023M4!KJ"
𝑌2 𝑓 = 𝑋2[𝑓]𝑒1234(K<6<QR)MJ"6 023M4! K<6<QR J"
Δ𝜙 = 2𝜋𝑓𝑁 + 𝐶𝑃 𝛿𝑇5
𝑁− 2𝜋𝛿𝑓0(𝑁 + 𝐶𝑃)𝑇5Phase accumulation:
• Residual CFO (Carrier Frequency Offset)
• Residual SFO (Sampling Frequency Offset)
Frequency Bins−𝑁/2 𝑁/2
y-intercept: CFO
Slope: SFO
Δ𝜙 = 2𝜋𝑓𝑁 + 𝐶𝑃 𝛿𝑇5
𝑁− 2𝜋𝛿𝑓0(𝑁 + 𝐶𝑃)𝑇5• Phase accumulation:
Phase Tracking• Residual CFO (Carrier Frequency Offset)
• Residual SFO (Sampling Frequency Offset)
OFDM Phase Tracking
Frequency Bins−𝑁/2 𝑁/2
y-intercept: CFO
Slope: SFO
• Sufficient to estimate slope & y-intercept to know the phase accumulated for all subcarriers.
• Use only few subcarriers as pilots & send known bits in them.
Δ𝜙 = 2𝜋𝑓𝑁 + 𝐶𝑃 𝛿𝑇5
𝑁− 2𝜋𝛿𝑓0(𝑁 + 𝐶𝑃)𝑇5• Phase accumulation:
OFDM Symbol
0−𝑁2
𝑁2− 1
Guard Bins
Guard Bins
DCPilots Pilots
−𝑁/2 𝑁/2
Δ𝜙Use Linear Regression to estimate phase accumulated
OFDM: Putting it Together
At TX:
• Create preamble symbol from training sequence (Uses BPSK)
• Repeat preamble symbol:
o 4 times for packet detectiono 2 times for CFO estimationo 2 times for channel estimationo Add CP for the last preamble
• Create data symbol from: o Data bits (Uses BPSK, QPSK, M-QAM)o Pilot bits (Uses BPSK)
• Add cyclic prefix to data symbols.
OFDM: Putting it Together
At RX:
• Detect beginning of packet.• Estimate & correct for CFO.• Jump ≈ 0.75 𝐶𝑃 samples into symbol to avoid ISI• Estimate the channel. • For each subsequent data symbol:
o Remove CPo Take FFT of Size No Correct for channelo Use linear regression to estimate residual CFO and SFOo Estimate accumulated phase Δ𝜙 𝑓 for each frequency bino Add Δ𝜙 𝑓 to channel estimate X𝐻 𝑓o Decode Bits
Progress so far?
A. I understand and I can follow
B. I understand most stuff but not everything
C. I can follow but I do not understand everything
D. I cannot follow and understood nothing about OFDM