channel capacity
• A very important consideration in data
communications is how fast we can
send data, in bits per second, over a
channel.
1
DATA RATE LIMITS
• The maximum data rate limit over a medium is
decided by following factors:
1. Bandwidth of channel.
2. Signal levels
3. Channel quality (level of noise)
• Two theoretical formulas were
developed to calculate the data rate:
one by Nyquist for a noiseless channel,
another by Shannon for a noisy
channel.
1. For noiseless channel- Nyquist bit rate
2. For noisy channel- Shannon capacity.
2
Nyquist Theorem
• Nyquist theorem states that “to produce the
original analog signal ,the sample rate must be
at least twice the highest frequency in the
original signal”.
Nyquist rate=2
˟
fmax
Nyquist rate is also called as Nyquist sample
rate.
• Sampling rate is inverse of sampling interval.
Sampling rate or sampling frequency.
fs = 1/TS .
3
Noiseless Channel : Nyquist bit rate
• Nyquist bit rate defines the theoretical maximum
bit rate for a noiseless channel or ideal channel.
• The formula for maximum bit rate in bits per
second(bps) is:
˟
˟
Where, BW =bandwidth at channel
Maximum bit rate = 2 BW log2L
L= number of signed levels used to
data.
represent
4
Noisy Channel : Shannon capacity
An ideal noiseless channel never exists. The maximum data
rate for any noisy channel is:
˟
C = BW log2 (1+S/N)
Where,
C= Channel capacity in bits per second
BW= bandwidth of channel
S/N= signal to noise ratio.
The channel capacity is also called as Shannon capacity. The
channel capacity do not depend upon the signal levels used to
represent the data.
5
Decibel
• The decibel (dB) measures the relative
strength of two signals or one signal at two
different points
• The decibel is positive if signal is strengthened
& it is negative when signal attenuates.
it is calculated as
db = 10 log10 (P2/P1)
• Variables P1 & P2 are the power of signal at
two points.
6
Noiseless Channel: Nyquist Rate
For a noiseless channel, the Nyquist bit rate formula
defines the theoretical maximum bit rate.
7
Example 3.33
Does the Nyquist theorem bit rate agree with the intuitive bit
rate described in baseband transmission?
Solution
They match when we have only two levels. We said, in
baseband transmission, the bit rate is 2 times the bandwidth
if we use only the first harmonic in the worst case. However,
the Nyquist formula is more general than what we derived
intuitively; it can be applied to baseband transmission and
modulation. Also, it can be applied when we have two or
more levels of signals.
8
Example
Consider a noiseless channel with a bandwidth of 3000 Hz
transmitting a signal with two signal levels. The maximum
bit rate can be calculated as
9
Example
Consider the same noiseless channel transmitting a signal
with four signal levels (for each level, we send 2 bits). The
maximum bit rate can be calculated as
10
Example
We need to send 265 kbps over a noiseless channel with a
bandwidth of 20 kHz. How many signal levels do we need?
Solution
We can use the Nyquist formula as shown:
Since this result is not a power of 2, we need to either
increase the number of levels or reduce the bit rate. If we
have 128 levels, the bit rate is 280 kbps. If we have 64
levels, the bit rate is 240 kbps.
11
Noisy Channel: Shannon Capacity
In reality, we cannot have a noiseless channel; the
channel is always noisy. In 1944, Claude Shannon
introduced a formula, called the Shannon capacity, to
determine the theoretical highest data rate for a noisy
channel:
12
Example
Consider an extremely noisy channel in which the value of
the signal-to-noise ratio is almost zero. In other words, the
noise is so strong that the signal is faint. What is the
Capacity C of the channel?
This means that the capacity of this channel is zero
regardless of the bandwidth. In other
words, we cannot receive any data through this channel.
13
Example
We can calculate the theoretical highest bit rate of a regular
telephone line. A telephone line normally has a bandwidth of
3000 Hz (300 to 3300 Hz) assigned for data
communications. The signal-to-noise ratio is usually 3162.
For this channel the capacity is calculated as
This means that the highest bit rate for a telephone line is
34.860 kbps. If we want to send data faster than this, we can
either increase the bandwidth of the line or improve the
signal-to-noise ratio.
3.14
Calculation of SNR in db
• The signal to noise ratio is often
given in decibels also.
• SNRdb = 10 log10 SNR
or
• SNRdb = 10 log10 S/N
15
Example
Assume that SNRdB = 36 and the channel bandwidth is 2
MHz. Calculate the theoretical channel capacity?
Assume that SNRdB = 45 and the channel bandwidth is 10
MHz. Calculate the theoretical channel capacity?
17
Using Both Limits
In practice, we need to use both methods to find the
limits and signal levels. Let us show this with an
example.
3.18
Example
We have a channel with a 1-MHz bandwidth. The SNR for
this channel is 63. What are the appropriate bit rate and
signal level?
Solution
First, we use the Shannon formula to find the upper limit.
The Shannon formula gives us 6 Mbps, the upper limit. For
better performance we choose something lower, 4 Mbps.
Then we use the Nyquist formula to find the number of
signal levels.
Calculate
3.19
L for 6
mbps
also.
Some More Examples
Example : Determine the data rate for a noiseless channel
having BW of 3kHz and two signal levels are used for
signal transmission.
Solution:
For a noiseless channel, the maximum data rate is given
by Nyquist bit rate as
˟ ˟
Bit rate = 2 (3 10 ) log (2)
˟ ˟ ˟
Bit rate = 6 10 log (2) / log
˟
˟
Bit rate = 6000bps
Bit rate= 2 BW log2L
3
3
2
10
2
(2)
20
Example : Calculate the BW of a noiseless channel having
maximum bit rate of 12kbps and four signal
levels.
Solution:
Nyquist bit rate is given by
˟ ˟
Bit rate = 2 BW log2L
˟
12 10 = 2 BW [log (4)]
˟
˟ ˟
BW = 12
10 / 2[log (4)/log
˟
BW = 3000Hz = 3kHz.
Given, Bit rate = 12
103 bps & L= 4
3
2
3
10
10
(2)]
21
Example: Compute the channel capacity of a
noisy channel having BW = 4Khz, & S/N =0.
Solution:
˟
C= BW Log2 (1+S/N)
˟
C=0
C=(4 103 )Log2 (1+0)
i.e Channel capacity is zero hence channel is
not able to transmit data.
22
Ex4. Calculate the capacity of a telephone channel.
The channel BW is 3000HZ & S/N is 3162.
Solution:
The telephone channel is a noisy channel
˟
C= 3000 log (1+3162)
˟
C= = 3000 (log 3162/log
˟
C= 34,860 bps
C=BW
log2 (1+S/N)
2
10
10
2)
23
Ex5. A system sends a signal that can assume 8
different voltage levels. it sends 400 of these
signals per second, what is baud rate?
Solution:
Baud rate is defined as number of signals
(symbols)transmitted per second since 400
symbols are transmitted per second.
Baud rate = 400 symbols/sec
24
Ex6. A system sends a signals that can assume 2
different voltage levels. It sends 100 of signals per
seconds, what is baud rate?
Solution:
Baud rate = number of symbols/sec
Bund rate = 100 symbols /sec.
25
Ex7. calculate the maximum bit rate for a channel
having bandwidth 3100Hz and SNR ratio 20 Db.
Solution:
We have to use formula (SNRdb = 10 log10 SNR)
Given
Given BW =3100Hz & SNR ratio =20 Db
i.e. 20 Db =10log[SNR],
SNR = 100
Maximum bit rate for noisy channel is given by
˟
= 3100 Log
˟
C=BW Log2 (1+S/N)
2
(1+100)
= 20,640 bits/sec
26
Ex8. Calculate the maximum bit rate for a
channel having bandwidth 3100Hz and S/N
ratio 10db.
Solution:
BW=3100Hz
S/N =10dB
10dB =10 LOG[S/N]
S/N = 10
˟
C=3100 [log
˟
C=BW
log2 (1+S/N)=3100
10
˟
log2 (1+10)
(11)/log10 2)=10,724bits/sec.
27
Ex9. Calculate the maximum bit rate for a channel having
bandwidth =1600Hz if
(a) S/N ratio 0 dB (b) S/N ratio 20 dB
Solution:
Given BW=1600Hz
a)
S/N = 0dB
0 = 10 LOG(S/N)
S/N =1
Maximum bit rate is given by
˟
˟
C=BW LOG2 (1+S/N)=1600 log2 (1+1)
C= 1600
˟
[log10 (2)/log10 (2)]=1600bits/sec
28
b)
S/N =20 dB
i.e. 20dB = 10log(S/N)
S/N =100
˟
= 1600 [log
˟
C =1600
log2 (1+100)
10
(101)/log10(2)]
C= 10,654 bits/sec
29
Example
Q. Suppose a signal travels through a transmission medium
and its power is reduced to one half. Calculate signal power
in db.
Ans: Since power is reduced to half this means that P2 = 0.5
P1. Hence, the attenuation (loss of power) can be calculated
as
A loss of 3 dB (−3 dB) is equivalent to losing one-half the
power.
3.30
Example
A signal travels through an amplifier, and its power is
increased 10 times. This means that P2 = 10P1. In this case,
the amplification (gain of power) can be calculated as
3.31
Figure: Decibels for given example
3.32
Example
One reason that engineers use the decibel to measure the
changes in the strength of a signal is that decibel numbers
can be added (or subtracted) when we are measuring several
points (cascading) instead of just two. In Figure a signal
travels from point 1 to point 4. The signal is attenuated by
the time it reaches point 2. Between points 2 and 3, the
signal is amplified. Again, between points 3 and 4, the signal
is attenuated. We can find the resultant decibel value for the
signal just by adding the decibel measurements between
each set of points. In this case, the decibel value can be
calculated as
3.33
Example
Sometimes the decibel is used to measure signal power in
milliwatts. In this case, it is referred to as dBm and is
calculated as dBm = 10 log10 Pm, where Pm is the power in
milliwatts.
Q. Calculate the power of a signal if its dBm = −30.
Solution
We can calculate the power in the signal as
dBm = 10 log10 Pm
=> -30 = 10 log10 Pm
=> log10 Pm = -3
=> Pm = 10-3 mW
Example
The loss in a cable is usually defined in decibels per
kilometer (dB/km). If the signal at the beginning of a cable
with −0.3 dB/km has a power of 2 mW, what is the power of
the signal at 5 km?
Solution
The loss in the cable in decibels is 5 × (−0.3) = −1.5 dB. We
can calculate the power as
3.35