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Lecture 4: Capacity
Project info 1. Each project group consists of two students.
2. Each project group should as soon as possible, send an email to [email protected] and
[email protected] containing Name and email address to each project member.’
3. The project group should contact Fredrik Rusek to decide about project and articles!
4. Each group should write a project report.
5. The structure of the project report should follow journal articles published by IEEE. However,
two columns are not needed.
6. The project report should be written in English with your own words, tables and figures, and
contain 4-5 pages.
7. The report should be clearly written, and written to the other students in this course!
8. Observe copyright rules: ”copy & paste” is in general strictly forbidden!
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Lecture 4: Capacity
Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested in
discussing how a good report should be written (book meetings with him via email). You can
also discuss topics and articles with Mgeni
10. The project report should be sent in .pdf format to Mgeni before Wednesday 12 December,
17.00
11. Feedback on the reports will be provided in a meeting with Mgeni (book meetings with him via
email)
12. Oral presentations in the week starting with Monday December 17
13. Each group should have relevant comments and questions on the project report and on the
oral presentation of another group. NOTE! The project presentation should be clear and aimed
to the other students in this course! After the oral presentation the project report and the
presentation will be discussed (5 min).
14. Final report should be sent to Fredrik and Mgeni at latest January 11, 2019.
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Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that
![Page 4: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/4.jpg)
Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies
![Page 5: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/5.jpg)
Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies
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Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus,
![Page 7: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/7.jpg)
Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently,
![Page 8: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/8.jpg)
Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
![Page 9: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/9.jpg)
Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
We have seen this before, it is defined as…
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Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
We have seen this before, it is defined as bandwidth efficiency
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Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
We have seen this before, it is defined as bandwidth efficiency
Power efficiency
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Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
We have seen this before, it is defined as bandwidth efficiency
Performance req
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Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
Bandwidth and power efficiencies are linked
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Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
Unit ?
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Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
Power
W
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Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
Bandwidth
W
Hz
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Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
Spectral density
W
Hz ?
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Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
Spectral density
W
Hz W/Hz
![Page 19: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/19.jpg)
Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
Has no unit (dimensionless)
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Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
Dito
![Page 21: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/21.jpg)
Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
Dito
![Page 22: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/22.jpg)
Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
Received signal-to-noise-power-ratio
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Lecture 4: Capacity
Power efficiency We know from before (e.g., union bound) that To meet a specific error probability target, this implies We also know that the transmit power satisfies Thus, or, equivalently, Now, divide both sides with the bandwidth W
Definition
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Lecture 4: Capacity
Power efficiency
”BW efficiency” = ”Signal-to-noise-power-ratio” x ”Power efficiency”
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Lecture 4: Capacity
Shannon Capacity
Before going on, we go through what the term capacity means
Given a scalar channel of form
We know that the capacity is
But what does this mean?
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Build a codebook of all information sequences possible to send of length K
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Lecture 4: Capacity
Shannon Capacity
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Build a codebook of all information sequences possible to send of length K
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Sending K bits of information means: pick one of the rows, and tell the receiver
which row it is
Lecture 4: Capacity
Shannon Capacity
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Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Build a codebook of codewords to send for each information word, length N
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
Lecture 4: Capacity
Shannon Capacity
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Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
Lecture 4: Capacity
Shannon Capacity
![Page 30: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/30.jpg)
Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
If this is my data
Lecture 4: Capacity
Shannon Capacity
![Page 31: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/31.jpg)
Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
If this is my data I send this one
Lecture 4: Capacity
Shannon Capacity
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Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
If this is my data
As x over this channel used N times
Lecture 4: Capacity
Shannon Capacity
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Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
Clearly, bit rate is K/N bits/channel use
Lecture 4: Capacity
Shannon Capacity
![Page 34: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/34.jpg)
Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
y1y2y3y4 ….. y(N-1)yN
Receiver observes
Lecture 4: Capacity
![Page 35: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/35.jpg)
Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
y1y2y3y4 ….. y(N-1)yN
Receiver observes
Compare with this one
Lecture 4: Capacity
![Page 36: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/36.jpg)
Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
y1y2y3y4 ….. y(N-1)yN
Receiver observes
Compare with this one
Lecture 4: Capacity
![Page 37: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/37.jpg)
Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
y1y2y3y4 ….. y(N-1)yN
Receiver observes
Compare with this one
Lecture 4: Capacity
![Page 38: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/38.jpg)
Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
y1y2y3y4 ….. y(N-1)yN
Receiver observes
Take smallest
Lecture 4: Capacity
![Page 39: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/39.jpg)
Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
y1y2y3y4 ….. y(N-1)yN
Receiver observes
Take smallest
So data is this one
Lecture 4: Capacity
![Page 40: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/40.jpg)
Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
y1y2y3y4 ….. y(N-1)yN
Receiver observes
Take smallest
So data is this one
This is ML decoding and is optimal Capacity means the following
Lecture 4: Capacity
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Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
y1y2y3y4 ….. y(N-1)yN
Receiver observes
Take smallest
So data is this one
This is ML decoding and is optimal Capacity means the following 1. If K/N ≤ C, and K->∞ then Prob(Correct detection)=1
Lecture 4: Capacity
![Page 42: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/42.jpg)
Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
y1y2y3y4 ….. y(N-1)yN
Receiver observes
Take smallest
So data is this one
This is ML decoding and is optimal Capacity means the following 1. If K/N ≤ C, and K->∞ then Prob(Correct detection)=1 2. If K/N > C, then Prob(Incorrect detection)=1
Lecture 4: Capacity
![Page 43: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/43.jpg)
Information book
000000 ….. 00 000000 ….. 01 000000 ….. 10
1111111 ….. 10 1111111 ….. 11
K
Codebook
x11x12x13x14 ….. x1(N-1)x1N
x21x22x23x24 ….. x2(N-1)x2N
x2 1x2 2x2 3x2 4 ….. x2 (N-1)x2 N K K K K K K
N
y1y2y3y4 ….. y(N-1)yN
Receiver observes
Take smallest
So data is this one
To reach C, code-symbols must be Random complex Gaussian variables That is, generate codebook randomly If it is generated with, say, 16QAM C cannot be reached
Lecture 4: Capacity
![Page 44: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/44.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
![Page 45: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/45.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Interpretation of capacity: Given a transmission of length T (seconds)
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
![Page 46: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/46.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Interpretation of capacity: Given a transmission of length T (seconds)
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
s(t)
T t
![Page 47: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/47.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Interpretation of capacity: Given a transmission of length T (seconds) And a number of bits K
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
s(t)
T t
Bits 0010111010110100...010011
K
![Page 48: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/48.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Interpretation of capacity: Given a transmission of length T (seconds) And a number of bits K The bitrate is: K/T [bit/sec]
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
s(t)
T t
Bits 0010111010110100...010011
K
![Page 49: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/49.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Interpretation of capacity: Given a transmission of length T (seconds) And a number of bits K The bitrate is: K/T [bit/sec] If K/T is too high, then many errors
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 11010..
s(t)
T t
Bits 0010111010110100...010011
K
![Page 50: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/50.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Interpretation of capacity: Given a transmission of length T (seconds) And a number of bits K The bitrate is: K/T [bit/sec] If K/T is too high, then many errors Shannon proved: Possible to have NO ERRORS if, 1) 2)
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
s(t)
T t
Bits 0010111010110100...010011
K
![Page 51: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/51.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Facts: 1. C is not power, nor bandwidth efficiency
(C is not dimensionless)
![Page 52: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/52.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Facts: 1. C is not power, nor bandwidth efficiency
(C is not dimensionless)
2. Not easy to reach C (i.e., to find a set of s(t) signals)
![Page 53: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/53.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Facts: 1. C is not power, nor bandwidth efficiency
(C is not dimensionless)
2. Not easy to reach C (i.e., to find a set of s(t) signals)
3. There is no parameter called
![Page 54: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/54.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Facts: 1. C is not power, nor bandwidth efficiency
(C is not dimensionless)
2. Not easy to reach C (i.e., to find a set of s(t) signals)
3. There is no parameter called
4. When W grows:
![Page 55: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/55.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Facts: 1. C is not power, nor bandwidth efficiency
(C is not dimensionless)
2. Not easy to reach C (i.e., to find a set of s(t) signals)
3. There is no parameter called
4. When W grows:
Grows linearly
![Page 56: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/56.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Facts: 1. C is not power, nor bandwidth efficiency
(C is not dimensionless)
2. Not easy to reach C (i.e., to find a set of s(t) signals)
3. There is no parameter called
4. When W grows:
Grows linearly decreases
![Page 57: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/57.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Facts: 1. C is not power, nor bandwidth efficiency
(C is not dimensionless)
2. Not easy to reach C (i.e., to find a set of s(t) signals)
3. There is no parameter called
4. When W grows:
Grows linearly
Decreases logarihtmically
![Page 58: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/58.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Facts: 1. C is not power, nor bandwidth efficiency
(C is not dimensionless)
2. Not easy to reach C (i.e., to find a set of s(t) signals)
3. There is no parameter called
4. When W grows: C grows
Grows
![Page 59: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/59.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Example, P/N0 =1 C
W
But it grows to a limit
![Page 60: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/60.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
What is the limit?
Standard limit
![Page 61: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/61.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
What is the limit?
Standard limit
Identify x with W
![Page 62: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/62.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
What is the limit?
Standard limit
Identify x with W
Identify A with P/N0
![Page 63: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/63.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
What is the limit?
Standard limit
Identify x with W
Identify A with P/N0 Express log2(x) as ln(x)/ln(2)
![Page 64: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/64.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
What is the limit?
Standard limit
Carry out limit
![Page 65: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/65.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
What is the limit?
Standard limit
Carry out limit
![Page 66: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/66.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
What is the limit?
Standard limit
Carry out limit
![Page 67: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/67.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Standard limit
C
W
But it grows to a limit
Example, P/N0 =1
![Page 68: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/68.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Standard limit
C
W
But it grows to a limit
Example, P/N0 =1
![Page 69: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/69.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Standard limit
C
W
But it grows to a limit
Example, P/N0 =1
![Page 70: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/70.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Summary 1. We stated that the capacity of the above is bits/second
2. We proved that for infinite bandwidth, the capacity is
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
![Page 71: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/71.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Summary 1. We stated that the capacity of the above is bits/second
2. We proved that for infinite bandwidth, the capacity is
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
![Page 72: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/72.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Bandwidth efficiency By definition,
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
![Page 73: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/73.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Bandwidth efficiency By definition, Effect of increasing/decreasing W ?
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
![Page 74: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/74.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Bandwidth efficiency By definition, Effect of increasing/decreasing W ? For large W, BW efficiency = 0 For small W, BW efficiency = ∞
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
![Page 75: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/75.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Bandwidth efficiency By definition, Effect of increasing/decreasing W ? For large W, BW efficiency = 0 For small W, BW efficiency = ∞
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
However
Bandwidth vs. Power efficiency
![Page 76: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/76.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Bandwidth efficiency By definition, Effect of increasing/decreasing W ? For large W, BW efficiency = 0 For small W, BW efficiency = ∞
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
However So,
Bandwidth vs. Power efficiency
![Page 77: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/77.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Bandwidth efficiency By definition, Effect of increasing/decreasing W ? For large W, BW efficiency = 0 For small W, BW efficiency = ∞
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
However So, Or, equivalently
Bandwidth vs. Power efficiency
![Page 78: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/78.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
However So, Or, equivalently
Bandwidth vs. Power efficiency
What happens if C/W grows?
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Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
However So, Or, equivalently
Bandwidth vs. Power efficiency
What happens if C/W grows? Eb/N0 grows as well In fact, we have (check at home) that to have 0 error probability
![Page 80: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/80.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
However So, Or, equivalently
Bandwidth vs. Power efficiency
What happens if C/W grows? Eb/N0 grows as well In fact, we have (check at home) that to have 0 error probability But, since Eb/N0 grows with C/W, there must be a minimum Eb/N0 achieved at vanishing C/W Standard limit:
![Page 81: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/81.jpg)
Lecture 4: Capacity
Extension to continuous channel (Shannon ’48) System model:
Transmitter
P [W] Bits 00101..
Channel
W f
1 s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
However So, Thus
Bandwidth vs. Power efficiency
What happens if C/W grows? Eb/N0 grows as well In fact, we have (check at home) that to have 0 error probability But, since Eb/N0 grows with C/W, there must be a minimum Eb/N0 achieved at vanishing C/W Standard limit:
![Page 82: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/82.jpg)
Lecture 4: Capacity
![Page 83: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/83.jpg)
Lecture 4: Capacity
Ps=0
![Page 84: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/84.jpg)
Lecture 4: Capacity
Eb/N0 grows if C/W grows coding
NOTE: Plot does not tell what the capacity is in bit/sec
Impossible region, since
Ps=0
![Page 85: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/85.jpg)
Lecture 4: Capacity
Ps=0
C/W -> 0
![Page 86: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/86.jpg)
Lecture 4: Capacity
Ps=0
C/W -> 0
Eb/N0 grows if C/W grows
![Page 87: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/87.jpg)
Lecture 4: Capacity
C/W -> 0
Eb/N0 grows if C/W grows
Uncoded systems, must be plotted at low error probability
![Page 88: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/88.jpg)
Lecture 4: Capacity
C/W -> 0
Eb/N0 grows if C/W grows
Uncoded systems, must be plotted at low error probability
coding
![Page 89: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/89.jpg)
Lecture 4: Capacity
C/W -> 0
Eb/N0 grows if C/W grows
Uncoded systems, must be plotted at low error probability
coding
NOTE: Plot does not tell what the capacity is in bit/sec
![Page 90: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/90.jpg)
Lecture 4: Capacity
Extension to frequency dependent channel System model:
Transmitter
P [W] Bits 00101..
Channel
W f
s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
![Page 91: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/91.jpg)
Lecture 4: Capacity
Extension to frequency dependent channel System model:
Transmitter
P [W] Bits 00101..
Channel
W f
s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Frequency response of channel
|H(f)|2
![Page 92: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/92.jpg)
Lecture 4: Capacity
Extension to frequency dependent channel System model:
Transmitter
P [W] Bits 00101..
Channel
W f
s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Frequency response of channel
Signal also have frequency representation
|H(f)|2
![Page 93: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/93.jpg)
Lecture 4: Capacity
Extension to frequency dependent channel System model:
Transmitter
P [W] Bits 00101..
Channel
W f
s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Frequency response of channel Power spectral density is what matters
R(f) |H(f)|2
![Page 94: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/94.jpg)
Lecture 4: Capacity
Extension to frequency dependent channel System model:
Transmitter
P [W] Bits 00101..
Channel
W f
s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Frequency response of channel Power spectral density is what matters
Constraint on R(f) ?
R(f) |H(f)|2
![Page 95: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/95.jpg)
Lecture 4: Capacity
Extension to frequency dependent channel System model:
Transmitter
P [W] Bits 00101..
Channel
W f
s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Frequency response of channel Power spectral density is what matters
Constraint on R(f) ?
R(f) |H(f)|2
![Page 96: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/96.jpg)
Lecture 4: Capacity
Extension to frequency dependent channel System model:
Transmitter
P [W] Bits 00101..
Channel
W f
s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Frequency response of channel Power spectral density is what matters
Constraint on R(f) ? If deep fade here
R(f) |H(f)|2
![Page 97: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/97.jpg)
Lecture 4: Capacity
Extension to frequency dependent channel System model:
Transmitter
P [W] Bits 00101..
Channel
W f
s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Frequency response of channel Power spectral density is what matters
Constraint on R(f) ? Stupidity to put power here
R(f) |H(f)|2
![Page 98: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/98.jpg)
Lecture 4: Capacity
Extension to frequency dependent channel System model:
Transmitter
P [W] Bits 00101..
Channel
W f
s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Frequency response of channel Power spectral density is what matters
Conclusion: We should optimize the left plot, for the given right plot
Constraint on left plot is
R(f) |H(f)|2
![Page 99: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/99.jpg)
Lecture 4: Capacity
Noise N(t)
r(t) Receiver
ML 00101..
Frequency response of channel Power spectral density is what matters
R(f)
Frequency response of Noise
RN(f)
|H(f)|2
![Page 100: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/100.jpg)
Lecture 4: Capacity
Transmitter
P [W] Bits 00101..
Channel
W f
s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Frequency response of channel Power spectral density is what matters
Conclusion: We should optimize the left plot, for the given right plot
Constraint on left plot is
R(f) |H(f)|2
Frequency response of Noise
RN(f)
![Page 101: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/101.jpg)
Lecture 4: Capacity
Transmitter
P [W] Bits 00101..
Channel
W f
s(t)
Noise N(t), N0
r(t) Receiver
ML 00101..
Frequency response of channel Power spectral density is what matters
Conclusion: We should optimize the left plot, for the given right plot
Constraint on left plot is
R(f) |H(f)|2
Frequency response of Noise
RN(f)
We need to find this formula
![Page 102: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/102.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
|H(f)|2 In this small piece We can use
![Page 103: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/103.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
|H(f)|2 In this small piece We can use
![Page 104: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/104.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
|H(f)|2 In this small piece We can use
For flat noise, RN(f) = N0/2
![Page 105: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/105.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
|H(f)|2 In this small piece We can use
For flat noise, RN(f) = N0/2
=2RN(f)
![Page 106: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/106.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
|H(f)|2 In this small piece We can use
![Page 107: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/107.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
|H(f)|2 In this small piece We can use
How much power do we have?
![Page 108: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/108.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
|H(f)|2 In this small piece We can use
How much power do we have?
![Page 109: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/109.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
|H(f)|2 In this small piece We can use
![Page 110: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/110.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
|H(f)|2 In this small piece We can use
![Page 111: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/111.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
In this small piece We can use Sum up
![Page 112: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/112.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
Sum up
![Page 113: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/113.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
![Page 114: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/114.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
How to solve the below problem? WATERFILLING
![Page 115: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/115.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
How to solve the below problem? WATERFILLING Step 1. Find and plot
![Page 116: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/116.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
How to solve the below problem? WATERFILLING Step 1. Find and plot Step 2. Fill a bucket with P units of water
![Page 117: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/117.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
How to solve the below problem? WATERFILLING Step 1. Find and plot Step 2. Fill a bucket with P units of water Step 3. Pour it in the shape
![Page 118: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/118.jpg)
Lecture 4: Capacity
R(f) |H(f)|2 RN(f)
How to solve the below problem? WATERFILLING Step 1. Find and plot Step 2. Fill a bucket with P units of water Step 3. Pour it in the shape Step 4. R(f) is the water-level
![Page 119: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/119.jpg)
Lecture 4: Capacity
How to solve the below problem? WATERFILLING Step 1. Find and plot Step 2. Fill a bucket with P units of water Step 3. Pour it in the shape Step 4. R(f) is the water-level
On Exam, |H(f)|^2 would be ”nice”, such as
![Page 120: Lecture 4: Capacity - Lunds tekniska högskola€¦ · Lecture 4: Capacity Project info 9. Book a meeting with Mgeni at your earliest possible convenience in case you are interested](https://reader036.vdocument.in/reader036/viewer/2022062601/5ed82e280fa3e705ec0dfc48/html5/thumbnails/120.jpg)
Lecture 4: Capacity
How to solve the below problem? WATERFILLING Step 1. Find and plot Step 2. Fill a bucket with P units of water Step 3. Pour it in the shape Step 4. R(f) is the water-level Observations:
1. Good channels get more power than bad 2. At very high SNRs, all channels get,
roughly, the same power