ECEN 463 – Wireless Communication
Background and Introduction to the wireless channel
Lecture 3
Dr. Mohammed Karmoose
Reminder
Passband vs baseband models
bits
Complex
pulse
shaper
Matched
Filter
𝑒 𝑗2𝜋𝑓𝑡 ~
Decision
Make
−𝑗2𝜋𝑓𝑡
𝑒
~
Complex passband model
Pros:
• Allows to study the behavior of
the system at the level of
waveforms
Cons:
• Requires heavy computations
due to the large sampling
frequency needed
bits
Reminder
Passband vs baseband models
bits
Mapper
bits
Demapper
Complex baseband model
10
Input bits
Symbol
00
-1-j
01
1-j
10
-1+j
11
1+j
11
1
-1
00
Mapping table
1
-1
Constellation diagram
01
Baseband model in Additive White Gaussian
Noise (AWGN)
4
Passband model in AWGN
𝑛(𝑡)
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
Matched
Filter
~
Decision
Make
bits
~
We usually aim to analyze the performance of communication systems in
the presence of additive white Gaussian noise.
Passband model in AWGN
𝑛(𝑡)
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
Matched
Filter
~
Decision
Make
bits
~
We usually aim to analyze the performance of communication systems in
the presence of additive white Gaussian noise.
The level of noise power/energy is usually determined compared to the
level of signal power/energy through the notion of Signal-to-Noise-Ratio
(SNR), or in case of digital signals, Eb/No and Es/No
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
𝑠𝐼 (𝑡)
𝐴
𝑇𝑠
-𝐴
𝑡
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
𝑠𝐼 (𝑡)
~
Average energy per symbol is given by
𝐴
𝑇𝑠
-𝐴
𝑡
𝑨𝟐 𝑻𝒔 + −𝑨 𝟐 𝑻𝒔
𝑬𝒔 =
×𝟐
𝟐
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
𝑠𝐼 (𝑡)
~
Average energy per symbol is given by
𝐴
𝑇𝑠
𝑡
𝑨𝟐 𝑻𝒔 + −𝑨 𝟐 𝑻𝒔
𝑬𝒔 =
×𝟐
𝟐
Energy in one
symbol
-𝐴
I and Q channels
Two symbols
Energy in the
other symbol
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
𝑠𝐼 (𝑡)
𝐴
𝑇𝑠
𝑡
𝑨𝟐 𝑻𝒔 + −𝑨 𝟐 𝑻𝒔
𝑬𝒔 =
×𝟐
𝟐
Energy in one
symbol
-𝐴
~
Task: How would 𝐸𝑠
changeAverage
for other
energy per symbol is given by
modulations?
I and Q channels
Two symbols
Energy in the
other symbol
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
Noise Power Spectral Density
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
Noise Power Spectral Density
𝑁𝑜 /2
𝑓
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
Noise Power Spectral Density
This is 𝑁𝑜 /2 and we use 𝑁𝑜 in
the Es/No expression
𝑁𝑜 /2
𝑓
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Eb/No: This is the “Average
Energy per Bit” over “Noise
Power Spectral Density”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Eb/No: This is the “Average
Energy per Bit” over “Noise
Power Spectral Density”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
𝐸𝑠 : Average energy per symbol
𝐸𝑏 : Average energy per bit
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Eb/No: This is the “Average
Energy per Bit” over “Noise
Power Spectral Density”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
𝐸𝑠 : Average energy per symbol
𝐸𝑏 : Average energy per bit
10
1
1
-1
00
11
-1
QPSK
01
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Eb/No: This is the “Average
Energy per Bit” over “Noise
Power Spectral Density”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
𝐸𝑠 : Average energy per symbol
𝐸𝑏 : Average energy per bit
10
1
11
Every symbol carriers 2 bits
1
-1
00
-1
QPSK
01
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Eb/No: This is the “Average
Energy per Bit” over “Noise
Power Spectral Density”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
𝐸𝑠 : Average energy per symbol
𝐸𝑏 : Average energy per bit
10
1
11
Every symbol carriers 2 bits
1
-1
∴ 𝐸𝑏 = 𝐸𝑠 /2
𝟏
∴ 𝑬𝒃 /𝑵𝒐 = 𝑬𝒔 /𝑵𝒐
𝟐
00
-1
QPSK
01
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Eb/No: This is the “Average
Energy per Bit” over “Noise
Power Spectral Density”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
𝐸𝑠 : Average energy per symbol
𝐸𝑏 : Average energy per bit
10
1
11
Every symbol carriers 2 bits
∴ 𝐸𝑏 = 𝐸𝑠 /2
𝟏
∴ 𝑬𝒃 /𝑵𝒐 = 𝑬𝒔 /𝑵𝒐
𝟐
~
Task: What is relation
1
-1
between 𝐸𝑏 and 𝐸𝑠 for other
modulations?
00
-1
QPSK
01
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
SNR: This is the “Average
Signal Power” over “Noise
Power”
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
SNR: This is the “Average
Signal Power” over “Noise
Power”
𝑃
𝑆𝑁𝑅 = 2
𝜎𝑛
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
SNR: This is the “Average
Signal Power” over “Noise
Power”
𝑃
𝑆𝑁𝑅 = 2
𝜎𝑛
𝑃: Average signal power
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
SNR: This is the “Average
Signal Power” over “Noise
Power”
Complex
pulse
shaper
bits
𝑃
𝑆𝑁𝑅 = 2
𝜎𝑛
𝑃: Average signal power
𝑃 = 𝐸𝑠 /𝑇𝑠
𝑒 𝑗2𝜋𝑓𝑡
𝑠𝐼 (𝑡)
𝐴
𝑇𝑠
-𝐴
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
SNR: This is the “Average
Signal Power” over “Noise
Power”
𝑃
𝑆𝑁𝑅 = 2
𝜎𝑛
𝑃: Average signal power
𝑃 = 𝐸𝑠 /𝑇𝑠
𝜎𝑛2 : Noise power
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
SNR: This is the “Average
Signal Power” over “Noise
Power”
bits
Complex
pulse
shaper
𝑃
𝑆𝑁𝑅 = 2
𝜎𝑛
𝑒 𝑗2𝜋𝑓𝑡
~
𝑃: Average signal power
𝑃 = 𝐸𝑠 /𝑇𝑠
𝜎𝑛2 : Noise power
𝜎𝑛2 = 𝑁𝑜 × 𝐵𝑊 = 𝑁𝑜 /𝑇𝑠
𝐵𝑊
𝐵𝑊
𝑁𝑜 /2
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
SNR: This is the “Average
Signal Power” over “Noise
Power”
bits
Complex
pulse
shaper
𝑃
𝑆𝑁𝑅 = 2
𝜎𝑛
𝑒 𝑗2𝜋𝑓𝑡
𝑃: Average signal power
𝑃 = 𝐸𝑠 /𝑇𝑠
𝜎𝑛2 : Noise power
𝜎𝑛2 = 𝑁𝑜 × 𝐵𝑊 = 𝑁𝑜 /𝑇𝑠
𝑷
𝐄𝐬
∴ 𝑺𝑵𝑹 = 𝟐 =
𝝈𝒏 𝐍𝐨
~
Passband model in AWGN
𝑛(𝑡)
Definitions of SNR, Eb/No and Es/No:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
𝟐
bits
𝑒 𝑗2𝜋𝑓𝑡
𝟐
𝑨 𝑻𝒔 + −𝑨 𝑻𝒔
𝑬𝒔 =
×𝟐
𝟐
•
Eb/No: This is the “Average
Energy per Bit” over “Noise
Power Spectral Density”
𝟏
∴ 𝑬𝒃 /𝑵𝒐 = 𝑬𝒔 /𝑵𝒐
𝟐
Complex
pulse
shaper
•
~
SNR: This is the “Average Signal
Power” over “Noise Power”
𝑷
𝐄𝐬
∴ 𝑺𝑵𝑹 = 𝟐 =
𝝈𝒏 𝐍𝐨
Passband model in AWGN
𝑛(𝑡)
Noise modeling in the passband
model:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
bits
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
𝑦(𝑡)
~
Passband model in AWGN
𝑛(𝑡)
Noise modeling in the passband
model:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
bits
𝑦 𝑡 = 𝑠𝐼 𝑡 + 𝑗 𝑠𝑄 𝑡
𝑒 𝑗2𝜋𝑓𝑡 + 𝑛(𝑡)
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
𝑦(𝑡)
~
Passband model in AWGN
𝑛(𝑡)
Noise modeling in the passband
model:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
bits
𝑦 𝑡 = 𝑠𝐼 𝑡 + 𝑗 𝑠𝑄 𝑡
𝑃
𝑒 𝑗2𝜋𝑓𝑡 + 𝑛(𝑡)
𝜎𝑛2
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
𝑦(𝑡)
~
Passband model in AWGN
𝑛(𝑡)
Noise modeling in the passband
model:
𝑠𝐼 𝑡 + 𝑗𝑠𝑄 (𝑡)
bits
𝑦 𝑡 = 𝑠𝐼 𝑡 + 𝑗 𝑠𝑄 𝑡
𝑃
𝑒 𝑗2𝜋𝑓𝑡 + 𝑛(𝑡)
Complex
pulse
shaper
𝑒 𝑗2𝜋𝑓𝑡
𝜎𝑛2
𝑛 𝑡 = 𝑛𝐼 𝑡 + 𝑗 𝑛𝑄 (𝑡)
Gaussian
random
process
𝜎𝑛2
𝒩(0, )
2
Gaussian
random
process
𝜎𝑛2
𝒩(0, )
2
𝑦(𝑡)
~
Baseband model in AWGN
𝑤[𝑛]
Definitions of SNR, Eb/No and Es/No:
bits
𝑏[𝑛]
Mapper
𝑠[𝑛]
𝑦[𝑛]
Baseband model in AWGN
𝑤[𝑛]
Definitions of SNR, Eb/No and Es/No:
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
𝑏[𝑛]
Mapper
𝑠[𝑛]
𝑦[𝑛]
Baseband model in AWGN
𝑤[𝑛]
Definitions of SNR, Eb/No and Es/No:
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
𝑏[𝑛]
𝑠[𝑛]
Mapper
𝑦[𝑛]
10
a
11
𝐸𝑠 : Average energy per symbol
a
-a
00
-a
QPSK
01
Baseband model in AWGN
𝑤[𝑛]
Definitions of SNR, Eb/No and Es/No:
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
𝑏[𝑛]
𝑠[𝑛]
Mapper
𝑦[𝑛]
10
a
11
𝐸𝑠 : Average energy per symbol
If 𝑠[𝑛] is a symbol
a
-a
00
-a
QPSK
01
Baseband model in AWGN
𝑤[𝑛]
Definitions of SNR, Eb/No and Es/No:
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
𝑏[𝑛]
𝑠[𝑛]
Mapper
𝑦[𝑛]
10
a
11
𝐸𝑠 : Average energy per symbol
If 𝑠[𝑛] is a symbol → Its energy is 𝑠 𝑛 2
a
-a
00
-a
QPSK
01
Baseband model in AWGN
𝑤[𝑛]
Definitions of SNR, Eb/No and Es/No:
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
𝑏[𝑛]
𝑠[𝑛]
Mapper
𝑦[𝑛]
10
a
11
𝐸𝑠 : Average energy per symbol
If 𝑠[𝑛] is a symbol → Its energy is 𝑠 𝑛 2
a
-a
2
2
2
𝑓𝑜𝑟 00
2
2
2
𝑓𝑜𝑟 01
2
2
2
𝑓𝑜𝑟 10
2
2
2
𝑓𝑜𝑟 11
𝑎 + 𝑎 = 2𝑎
Energy in different
modulation symbols
𝑎 + 𝑎 = 2𝑎
𝑎 + 𝑎 = 2𝑎
𝑎 + 𝑎 = 2𝑎
00
-a
QPSK
01
Baseband model in AWGN
𝑤[𝑛]
Definitions of SNR, Eb/No and Es/No:
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
𝑏[𝑛]
𝑠[𝑛]
Mapper
𝑦[𝑛]
10
a
11
𝐸𝑠 : Average energy per symbol
If 𝑠[𝑛] is a symbol → Its energy is 𝑠 𝑛 2
a
-a
2
2
2
𝑓𝑜𝑟 00
2
2
2
𝑓𝑜𝑟 01
2
2
2
𝑓𝑜𝑟 10
2
2
2
𝑓𝑜𝑟 11
𝑎 + 𝑎 = 2𝑎
Energy in different
modulation symbols
𝑎 + 𝑎 = 2𝑎
𝑎 + 𝑎 = 2𝑎
𝑎 + 𝑎 = 2𝑎
𝟐𝒂𝟐 + 𝟐𝒂𝟐 + 𝟐𝒂𝟐 + 𝟐𝒂𝟐
∴ 𝑬𝒔 =
= 𝟐𝒂𝟐
𝟒
00
-a
QPSK
01
Baseband model in AWGN
𝑤[𝑛]
Definitions of SNR, Eb/No and Es/No:
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
𝑏[𝑛]
Mapper
𝑠[𝑛]
𝑦[𝑛]
𝐸𝑠
2
10
11
𝐸𝑠 : Average energy per symbol
If 𝑠[𝑛] is a symbol → Its energy is 𝑠 𝑛 2
2
2
2
𝑓𝑜𝑟 00
2
2
2
𝑓𝑜𝑟 01
2
2
2
𝑓𝑜𝑟 10
2
2
2
𝑓𝑜𝑟 11
𝑎 + 𝑎 = 2𝑎
Energy in different
modulation symbols
𝑎 + 𝑎 = 2𝑎
𝑎 + 𝑎 = 2𝑎
𝑎 + 𝑎 = 2𝑎
𝟐𝒂𝟐 + 𝟐𝒂𝟐 + 𝟐𝒂𝟐 + 𝟐𝒂𝟐
∴ 𝑬𝒔 =
= 𝟐𝒂𝟐
𝟒
−
𝐸𝑠
2
𝐸𝑠
2
00
−
QPSK
𝐸𝑠
2
01
Baseband model in AWGN
𝑤[𝑛]
Definitions of SNR, Eb/No and Es/No:
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
𝑏[𝑛]
Mapper
𝑠[𝑛]
𝑦[𝑛]
𝐸𝑠
2
10
11
𝐸𝑠 : Average energy per symbol
𝐸
𝐸𝑠
2
𝑠
If 𝑠[𝑛] is a symbol Task:
→ Its energy
𝑛 2relation
−
whatisis 𝑠the
2
between
𝐸 and the symbol
2
2 𝑠 2
𝑎 + 𝑎 = 2𝑎
𝑓𝑜𝑟 00
magnitude
2
2
2 for different
𝑎 + 𝑎 = 2𝑎
𝑓𝑜𝑟 01
Energy in different
2
2
2
𝑎 +modulations?
𝑎 = 2𝑎
𝑓𝑜𝑟 10
modulation symbols
2
2
𝑎 + 𝑎 = 2𝑎
2
𝑓𝑜𝑟 11
𝟐𝒂𝟐 + 𝟐𝒂𝟐 + 𝟐𝒂𝟐 + 𝟐𝒂𝟐
∴ 𝑬𝒔 =
= 𝟐𝒂𝟐
𝟒
00
−
QPSK
𝐸𝑠
2
01
Baseband model in AWGN
𝑤[𝑛]
Definitions of SNR, Eb/No and Es/No:
•
Es/No: This is the “Average
Energy per Symbol” over
“Noise Power Spectral
Density”
bits
𝑏[𝑛]
Mapper
𝑠[𝑛]
𝑦[𝑛]
𝐸𝑠
2
10
•
Eb/No: This is the “Average
Energy per bit” over “Noise
Power Spectral Density”
−
11
𝐸𝑠
2
𝐸𝑠
2
𝑬𝒃 𝟏 𝑬𝒔
=
𝑵𝒐 𝟐 𝑵𝒐
•
SNR: This is the “Average
signal power” over “noise
power”
𝑷
𝑬𝒔
𝑺𝑵𝑹 = 𝟐 =
𝝈𝒏 𝑵𝒐
00
−
QPSK
𝐸𝑠
2
01
Baseband model in AWGN
𝑤[𝑛]
Noise modeling in the baseband
model:
𝐸𝑠
𝑁𝑜
bits
𝑏[𝑛]
Mapper
𝑠[𝑛]
𝑦[𝑛]
𝑦 𝑛 = 𝑠 𝑛 + 𝑤[𝑛]
Baseband model in AWGN
𝑤[𝑛]
Noise modeling in the baseband
model:
𝐸𝑠
𝑁𝑜
bits
𝑏[𝑛]
Mapper
𝑠[𝑛]
𝑦[𝑛]
𝑦 𝑛 = 𝑠 𝑛 + 𝑤[𝑛]
𝑤[𝑛] = 𝑤𝐼 [𝑛] + 𝑗 𝑤𝑄 [𝑛]
Gaussian
random
variable
𝑁𝑜
𝒩(0, )
2
Gaussian
random
variable
𝑁𝑜
𝒩(0, )
2
Baseband model in AWGN
𝑤[𝑛]
Noise modeling in the baseband
model:
𝐸𝑠
𝑁𝑜
bits
𝑏[𝑛]
Mapper
𝑠[𝑛]
𝑦[𝑛]
𝑦 𝑛 = 𝑠 𝑛 + 𝑤[𝑛]
𝐸𝑠
2
10
𝑤[𝑛] = 𝑤𝐼 [𝑛] + 𝑗 𝑤𝑄 [𝑛]
−
Gaussian
random
variable
𝑁𝑜
𝒩(0, )
2
Gaussian
random
variable
𝑁𝑜
𝒩(0, )
2
11
𝐸𝑠
2
00
𝑠𝑛
𝐸𝑠
2
𝐸𝑠
−
2
QPSK
01
Baseband model in AWGN
𝑤[𝑛]
Noise modeling in the baseband
model:
𝐸𝑠
𝑁𝑜
bits
𝑏[𝑛]
𝑠[𝑛]
Mapper
𝑦[𝑛]
𝑦 𝑛 = 𝑠 𝑛 + 𝑤[𝑛]
𝐸𝑠
2
10
−
Gaussian
random
variable
𝑁𝑜
𝒩(0, )
2
𝑠𝑛
𝑤𝑛
𝑤[𝑛] = 𝑤𝐼 [𝑛] + 𝑗 𝑤𝑄 [𝑛]
Gaussian
random
variable
𝑁𝑜
𝒩(0, )
2
11
𝐸𝑠
2
00
𝐸𝑠
2
𝑦𝑛
𝐸𝑠
−
2
QPSK
01
Baseband model in AWGN
Modulation: QPSK, 𝑬𝒔 = 𝟏
𝑵𝟎 = 𝟏/𝟖
𝑵𝟎 = 𝟏/𝟒
Scatterplots at different values of 𝑬𝒔 /𝑵𝒐
𝑵𝟎 = 𝟏/𝟐
Baseband model in AWGN
Decoding symbols in baseband:
Decision region for 10 𝐸𝑠
2
10
−
𝐸𝑠
2
Decision region for 11
11
𝐸𝑠
2
Decision region for 00
Decision region for 01
𝐸𝑠
−
2
00
01
𝑦[𝑛]
Demapper
bits
Baseband model in AWGN
Decoding symbols in baseband:
Decision region for 10 𝐸𝑠
2
10
−
𝐸𝑠
2
Decision region for 11
11
𝐸𝑠
2
Decision region for 00
Task:
how would the
Decision region for 01
decision regions look like
for different modulations?
𝐸𝑠
−
2
00
𝑦[𝑛]
01
Demapper
bits