# Mod-01 Lec-10 Coherence Bandwidth of the Wireless Channel

Hello, welcome to this course on 3G 4G wireless
communications, before we start today’s lecture, let me begin with a recap of last lecture.
We said a multipath wireless channel can be represented as a sum of L multipath components. Each component is characterized by an attenuation
factor a and a delay tau that is the i th path has an attenuation a i and delay tau
i. We said the gain associated with the i th
path or the power associated with the i th path is magnitude a i square arriving at a
delay tau i. I can also represent magnitude a i square with g i the gain of the i th path or also the power associated
with the i th path, so that can be represented as a power g i as arriving at a delay tau
i. Then, we also said in a wireless channel we
have L paths the first path arriving at delay tau 0 second path delay tau 1 so on and so
forth. With the last path arriving a delay tau L minus 1 the first path has a power g 0 second path has a power g
1 so on and so forth the last path has a power g L minus 1. So, now my signal is arriving
over an interval of time unlike a wired communication channel where there is only one path because there
is no scattering in a wireless channel. Because of the scattering and the multipath propagation
environment I have multiple components arriving. These components are arriving over a certain
delay or a certain time interval this is also known as a time spread and this interval is
technically known as a delay spread of the signal, that spread of time over which the different signal copies
are arriving. We said this delay spread can be characterized by the quantity sigma tau
and we gave different expressions to compute sigma tau. We said one of the most popular ways to compute the delay
spread is what is known as the RMS or the root mean square delay spread. For that, we need to compute the average delay
which is tau bar tau bar is nothing but summation g i tau i divided by summation g i. So, if
I look at g i over summation g i it is nothing but the fractional power look at this g i is the power
in the i th path divided by summation of g i which is the total power. So, g i over summation
of g i is the fractional power this is the fractional power in the i th path, I am taking the fractional power
multiplying it by the delay. So, I am computing the weighted delay which is the average delay
tau bar. Then, I can compute my delay spread as the
fractional power times the deviation tau i minus tau bar square summation whole under
square root that gives me the average delay or the root mean square delay. We also did an example in which there are
four paths arriving with the different powers the first path is minus 20 dB minus 10 dB
and then so on so forth. We computed the average delay of this channel
or the RMS delay spread and we said the RMS delay spread value is 0.8573 micro seconds
and then we also said this power profile that we looked at varies from channel to channel. So, if we
have a large number of users in my 3 G, 4 G wireless cellular system, I can look at
all the channels and compute the average power profile. This is obtained by taking the expected value
of these power profiles or taking all these power profiles of different users, and computing
their average that is known as the average power profile which is phi bar of tau. Now, given phi bar of tau, I can compute the
average delay spread as follows, I compute f of tau which is the fractional power received
at the delay tau which is phi bar of tau divided by integral phi bar of tau. This is the total power the numerator
is the power at delay tau this gives me the fractional power at delay tau across all users
or on an average. The average RMS delay spread as f of tau which
is the fractional power times tau minus tau bar square which is the deviation integrated
from 0 to infinity whole under square root. This is similar to what we did in the RMS delay spread except
earlier expect that that was a discrete channel now we have a continuous power profile, so
we are using the integration. We also started with an example were I said
the average power profile is given as alpha e power minus tau over beta where alpha is
2 and beta is 1 microsecond. We said this is a standard exponential power profile in which the power
arriving at the receiver is decaying exponentially with delay. We were about to compute the power
the RMS delay spread of this exponential power profile. So, let me start today’s lecture
with a computation of the RMS delay spread of this power profile. We said phi bar of tau equals 2 e power minus
tau over beta were beta equals 1 micro second the fractional power profile in this case
f of tau is given as phi bar of tau divided by integral 0 to infinity phi bar of tau d tau. Let me first compute
this quantity phi bar of tau d tau phi bar of tau d tau integral 0 to infinity, this
is the total power. Remember, we said this is the total power this is nothing but 2 e power minus tau over beta integral 0 to
infinity which is nothing but 2 beta integral e power minus tau over beta 0 to infinity.
This is nothing but 2 beta hence f of tau is phi bar of tau over integral phi bar tau d tau which is nothing but phi
bar of tau divided by 2 beta. Hence, this can be derived as the fractional
power profile can be derived as 2 e power minus tau over beta divided by 2 beta equals
1 over beta e power minus tau over beta. So, this is the fractional power profile this is also the let me write
this down this is the fractional power profile this is the fraction of the power that is
received as the function of the delay tau, so this is the fractional power profile. Now, the average delay Tau bar is nothing
but tau bar equals tau f of tau d tau this is nothing but 0 to infinity tau over beta
e power minus tau over beta d tau. So, I am computing the average delay, I am saying that is tau times this is the fractional
power received. So, I am taking the delay weighing it by the fractional power integrating
from 0 to infinity that gives me the average delay that is nothing but integral tau over beta e power
minus tau over beta. This can be shown to be beta, so the average delay tau bar equals
beta equals 1 micro second, remember beta is a constant beta is a constant whose value is 1 micro second, now
let us compute the RMS delay spread. The RMS delay spread is nothing but sigma
tau square is given as integral 0 to infinity tau minus tau bar square f of tau d tau this
is nothing but integral 0 to infinity tau minus beta square 1 over beta e power minus tau over beta d tau. I will
not compute this integral explicitly over here, but you can compute it and you can verify
that the value of this integral is beta square. So, sigma tau square is beta square sigma tau is square root of beta
square which is beta which is equal to 1 microsecond hence what I have we derived. So far, what we have derived is for the exponential
power profile given as alpha e power minus tau over beta that is I have a power profile
which is exponential. So, the exponential power profile with alpha equals 2 beta equals 1 micro second,
we have derived the sigma tau or RMS delay spread
equals 1 micro second. So, we have derived
for an exponential profile that is when you average across all users in the cell if the power profile.
When the power profile looks like a decaying exponential that is as a function of the delay,
the arriving power at the receiver is decaying exponentially with the parameters alpha and beta.
That is the power profile is alpha e power minus tau over beta the RMS delay spread of
this wireless communication channel is nothing but beta which is 1 microsecond that is most
of the power is restricted to an interval or a delay spread
of 1 micro second. Now, with that intuition, let us go on to the next step which is characterizing
the delay spread of typical outdoor channels in 3 G, 4 G wireless systems. Let us come to the topic of average delay
spread in outdoor channels, let us consider a cell a typical cell with a base station
somewhere in the centre and a mobile somewhere at the of the edge of the cell. We know that typical cell radius is
around kilometers is around around 3 to 4, 5 kilometers sometimes 10 kilometers. Let
us call it around 2 kilometers, so the typical direct path is around 2 kilometers and let us say a scattered path
let us say there are scatters at the edge of the scale such as trees buildings and so
on and so forth. There is a scatter path that is coming towards
the mobile, so we are saying that there is we are considering a scenario in which there
is a cell phone at the boundary of the cell there is a base station at the centre of the cell. The direct path is
the radius of the cell which is around 2 to 3 kilometers which is the order of kilometers
there is a scatter path which is also of the order of kilometers which is let us say around 3 to 4 kilometers.
So, the if distance or the difference in the distance between the direct path and the scatter
path, so let me write it down clearly over here the difference in distance between scatter
path and direct path is approximately of the order of kilometers.
The direct path is around 2 kilometers the scatter path is around 3 to 4 kilometers,
so the distance the difference between these distances is around the order of kilometers which means the delay,
so let us say the direct path is arriving at tau 0. The direct path is arriving at tau 0 tau 0
is approximately equal to 2 kilometers over c tau delayed which is let us say tau one
which is the delayed path is arriving at approximately 2 to 3 kilometers over C. So, the delay spread which is the
difference between the arrival of the direct and scattered path the delay spread
or difference in time between direct. Scatter
path is approximately the difference in distance which is 2 minus 3
kilometers this distance is are of or the order of kilometers. So, the distances are
also of the order of the differences are also of the order of kilometers.
Let me say it is approximately 1 kilometer divided by C which is 1,000 meters divided
by 3 into 10 to the power of 8 meters per second which is c which is the velocity of
light or velocity of an electromagnetic wave in free space. This is
hence equal to 3.33 micro seconds, so what have we said so far, so far we have said if
you look at this we said the differences are these. The distance between the direct path and the scatter path
of the electromagnetic wave in the wireless cellular system or the transmitted radio wave
is approximately of the order of kilometers. This means the delay the spread of time over
which these signals are arising arriving is nothing but the difference in distance over
the velocity which is of the order of the difference. In this, distance is the of the order of kilometers over the velocity
which is 3 into 10 power 8 meters per second, this is 3.33 micro seconds. Now, I can say
the outdoor delay spreads in 3 G slash 4 G
wireless communication networks or wireless cellular networks or wireless communication
systems are approximately
of the order of micro second. Remember, the calculation that we
did earlier just in the previous page is not an exact calculation, it is an approximate
calculation, we said the distances are of the order of kilometers. Hence, the delay spreads are of the order of microseconds
typically around 1 to 3 micro seconds, so the outdoor delay spreads are approximately
around 1 to 3 micro seconds, let me take you to a standard values of delay spreads that are listed in
literature. For instance, let me go through this example
given in this slide over here. Here, different measurement campaigns have been carried out
to measure the delay spread of the arriving multipath components in wireless communication systems
at frequency nine ten mega hertz there have been delay spreads that have been reported.
Those are around an average delay spread of 1,300 nano seconds, nano second is 10 per minus 9 second,
each nano second is 1,000 micro seconds. So, this is approximately 1.3 micro seconds,
also you see in a worst case situation in San Francisco the delay spreads around the
worst case that is the maximum delay spread is around 10 to 25 micro seconds. You can also see different
values, for instance in this case average extreme case is around 1.9 to 2.1 micro seconds,
hence average outdoor delay spreads are around micro seconds, so let me go back to my lecture here. Let me say write that point again which is
average remember we are just talking about outdoor average outdoor that is for outside
cellular communication average outdoor delay spreads are around 1 to 2 micro seconds. Indoor delay spreads
are of course smaller because inside the distances are smaller the wall to wall distances the
wall to wall scattering distances are smaller. Therefore, the indoor delay spreads are much
smaller typically they are of the order of 10 to 20 or 10 to 15 nano 10 to 15 nano seconds.
So, indoor delay spreads for instance, if we are employing wireless LAN system inside a room that is
an a 2 dot 11 B system inside the room the indoor delay spreads because the distances
are smaller the indoor delay spreads are around 10 to 50 nano seconds. So, these are this is the typical
value of the indoor delay spread, now let me go on to another important idea related
to delay spread we have looked, so for at the delay spread.
We have said the delay spread is nothing but that interval of time over which signal copies
are arriving this happens because of the different multipath component the first component is
arriving second component so and so forth until the
L th component. So, these different components are arriving over an interval of time hence
there is a delay spread let us look at its implications in the frequency domain, so let us look at delay
spread in the frequency domain. That will lead us to an important idea known
as the coherence band width let us consider a channel delay profile h of tau which is
given which is given, so let us consider a channel delay profile. This is the channel delay profile, now what I want
to do is I will compute the Fourier transform and look at the spectrum of this delay profile.
So, let me compute the spectrum of this which is given as h of f equals 0 to infinity h of tau e power minus
j 2 pi f tau d tau. So, I am looking at the Fourier transform of this delay of this delay
profile which is h of tau or delay or this impulse response of the channel. Now, let me plot that Fourier transform let
us say it approximately looks like this alright so it is approximately constant for some bandwidth
around 0 then it starts following already it has a some kind of a low pass characteristic. So, it is flat,
so let us say this is the 0 frequency component it is flat for some width around 0 and this
it starts falling. Now, this portion or this bandwidth for which the frequency response, so this is I am plotting
magnitude of h of f. Now, this response or this portion over which
the response is approximately flat is known as the coherence bandwidth of the channel
and this is denoted by the symbol B c. So, what have we done so far we have taken a channel that is given
as a function of the delay. We have computed the Fourier transform of the channel delay
profile beside that it looks approximately like this that it has a low pass characteristic it is constant over some
frequency bandwidth. Then, it starts falling alright and this portion of the spectrum over
which the response is approximately constant is known as the coherence bandwidth.
You might also have seen low pass characteristics like this earlier and you might have characterized
this as things like the 3 dB bandwidth or null to null bandwidth and so on which is
essentially that region over. It has it has an approximately
flat response it is simply saying that region over which it allows signals to pass. So,
in this case we are being a bit more restrictive we are saying it is not only 3 dB, it has to be approximately flat and
this band width over which the response is flat of the response of the channel is flat
is known as the coherence bandwidth of the channel. This is an important idea in wireless communication, let me illustrate
you the relevance of the coherence bandwidth. We also know that if I transmit a signal to
a system which has frequency, let us say I transmit a signal with spectrum x of f through
a system which has a frequency response h of f, then the output response is simply x of f times h of f. I
am taking a signal x of f passing it through a filter whose response is h of f the output
has a frequency response x of f times h of f.
Now, let me take an example in our in our in our system this is the wireless channel
this h of f is the wireless channel x of f is the transmitted
and x of f h of f is nothing but the received
of the received signal let me write this as this is the spectrum of the received signal,
now let us look at this in the context of our coherence bandwidth. Let us say I have a wireless channel which
looks as follows this is my channel, this is its coherence band width this is the bandwidth
over which the frequency response is constant. I have a signal whose spectrum looks like this it is let me
draw the spectrum of the signal the signal has some spectrum, but the important thing
to note here is that the spectrum of the signal is less than the coherence band width. So, this is the coherence
bandwidth the signal spectrum is limited let me call this the bandwidth of the signal B
s is limited to the coherence band width that is the maximum frequency component less present in the signal
is less than B c over 2. So, the signal spectrum is limited to the
coherence band width, now at the output when I look at the output of this system when I
look at the output of this system. Since, this spectrum is limited to the coherence bandwidth over which the response
is flat I am multiplying some constant with this spectrum, hence at the output my spectrum
will be undistorted. Look at this, I am saying my signal this is my h of f which is my wireless channel
this is my x of f which is my transmitted signal.
This is my received signal h of f times x of f I am saying x of f has a bandwidth that
is less than the coherence bandwidth hence if I multiply h of f by x of f it is simply
getting multiplied by this constant which is the flat part or the gain of the
wireless channel. In the coherence bandwidth and hence the output spectrum the shape of
the output spectrum is same as the shape of the input spectrum, hence there is no distortion in other words
there is no distortion. So, what am I saying let me write this again
if signal bandwidth B s which is the signal band width less than or equal to B c which
is the coherence bandwidth. Then, there is no distortion no distortion in the received signal because the signal
bandwidth is less than the coherence bandwidth. There is no distortion in the received signal
because the signal spectrum is just getting multiplied by the constant part or the flat part of the band
width. Since there is no distortion whatever you transmit is a scale the received signal
is simply a scaled version of the transmitted signal. However, now consider the slightly different
case where I have some channel that looks like this were this is its coherence bandwidth
so this is my spectrum h of f and I have a signal whose bandwidth is much greater than the coherence
band width. This is my transmitted signal its band width, so this is the coherence bandwidth
of the channel which is the flat path over which the response is constant. I have a signal, but
my signal band width b of s is much larger than the coherence bandwidth B c, now when
I get the received signal which is H of f times X of f what happens?
This signal is multiplied by this spectrum in this part it is the flat part that is not
a problem however here it is significantly attenuated look at this here the response
is significantly different from the response in B c. So, this part of the signal
or this edges of the signal will be attenuated hence what I get which is this will look something
like this which means these parts, these portions. These are attenuated why because this signal has a band
width B signal which is greater than the coherence bandwidth.
So, I am taking my signal multiplying with the response of the channel in this part its
fine in the central part which is rests on the coherence band width it is multiplying
getting multiplied by the flat. Outside the coherence band width, there is
significant attenuation in the channel which means the edges of this signal are going to
get attenuated which means what I get is a distorted version of the input signal. Hence, if B s which is the signal bandwidth
is greater than B c which is the coherence band width then output or received signal
the received signal is distorted if my signal bandwidth is greater than the coherence bandwidth. Then, because outside
of the coherence bandwidth the channel is attenuating my received spectrum is attenuated
at the edges. This means my received spectrum is a distorted version of my transmitted spectrum
and hence coherence band width is an extremely important concept. Let me summarize that here if B s less than
B c no distortion the signal is essential restricted to the flat part of the channel
response. Hence, this is also known as a flat fading channel look at this my spectrum looks like this this is the coherence
bandwidth and my signal is restricted to this part in which it is the response is flat.
Here, the response is flat hence this is known as a flat fading channel however if B s is greater than B c. The signal
bandwidth is greater than greater than the channel coherence band width then there is
distortion
and the distortion is look at this distortion this is in my channel this is the coherence band width.
This is my signal with bandwidth larger than the signal the distortion in the central part
depends on the flat part the distortion in the edges depends on the frequency response
of the channel. Hence, this is also known as frequency selective
distortion which is when the bandwidth of the signal is greater than the coherence bandwidth
of the channel. This is H of f this is s this is X of f which is the transmitted signal.
The bandwidth of x of f is greater than h of f then the distortion the received spectrum
undergoes a distortion which depends on the frequency that is in this frequency. It depends
on the frequency response of the channel every frequency, it
depends on the frequency response of the channel. Hence, this is also known as frequency selective
distortion hence this is an important idea if the signal bandwidth is less than the coherence band
width. Then, there is no distortion if the and this
is a flat fading channel if the signal bandwidth is greater than the coherence band width.
Then, there is distortion and the distortion is frequency selective so it is known as frequency selective distortion
this is an important idea in 3G, 4G wireless communications hence I urge all of you again
to pay a close attention to this concept. So, let me summarize it for one last time
over here B s less than B c it is flat fading and B s greater than B c it is frequency selective
distortion. So, this is an important idea related to the coherence band width and the bandwidth of the signal
hence it is an important idea, hence please play close attention to this idea. Now, let us look at the frequency response
that can be explicitly computed from the delay profile we saw that h of t can be represent
as a i delta tau minus tau i. So, I can represent that delay profile as comprising of multiple paths i equals 0
to L minus 1 each path having an attenuation a y a i and corresponding to a delay that
is tau i. Now, if I compute the spectrum of this multi path delay profile that is h of f equals integral 0 to infinity
h of tau e power minus j 2 pi f tau d tau. I am computing the spectrum of this multipath
delay channel this is a multipath delay channel, I am computing the spectrum of this channel.
Now, this is integral 0 to infinity h of tau e power minus j to pi f tau d tau this is a standard Fourier
transform expression this is nothing but 0 to infinity.
Now, I am going to substitute the multipath delay profile over here that is i equals 0
to L minus 1 a i delta tau minus tau i e power minus j 2 pi f tau d tau. This can also be
represented by reversing by taking the x e power minus 2 pi j pi f tau
inside the summation that is i equals 0 to L minus 1 a i delta tau minus tau i e power
minus j 2 pi f tau. Now, you can observe that integral delta tau minus tau i e power minus j 2 pi f tau is nothing but
e power minus j 2 pi f tau i because integral f of t f of t delta t minus t naught is nothing
but f of t naught. So, this is nothing but e power minus j 2
pi f tau i
e power minus j 2 pi f tau i, this is the Fourier transform of the multipath channel.
So, this is the f t of the multipath 3 G 4 G wireless channel of the multipath this is the Fourier transform of
the multipath wireless channel. Here, I have sigma summation a i e power minus j 2 pi f
tau i a i is the attenuation of the i th path tau i is the delay of the i th path and summation i equals 0 to l minus 1
over the L paths this is the spectrum. Now, let me look at a typical path let me look
at a i e power minus j 2 pi f tau i. Let me look at a typical path at f equals
0 this is a i e power minus j two pi 0 times tau i which is nothing but e power 0 which
is 1, hence this is simply a i at f equals 1 over 4 tau i. This value is a i e power minus j 2 pi 1 over four tau i times
tau i equals a i e power minus j pi over 2 which is nothing but minus j over a i.
Now, look at these two quantities at f equals 0 this is a i at f equals 1 over 4 tau it
is 1 it is minus j i if a i is real at 1 over 4 tau i is has changed to an imaginary number
which is j times a i. So, which means from if f goes from 0 to 1 over 4 tau
i, it is changing completely as f goes from 0 to 1 over 4 tau i. The frequency response
is changing from a i to minus j i this is a real number this is an imaginary number. Essentially, what I am trying to say is if
you plot the frequency response of the complete channel as we said earlier it looks like this
it is constant for some duration and then it changes at some point at some point. It starts decaying what is
the point at which it starts decaying in other words what is the point at which this response
here is significantly different from 0. We can say that point let me call this point of significant change look
at this here the response is a i, here the response can be approximately has changed
dramatically. So, it is it has we are saying this is something
like j a i, so at this point the response has changed drastically compared to what the
response is at 0 the point of significant change is nothing but 1 over 4 t tau i. This is approximately what we are
saying is as we start moving from 0 frequency start increasing the deviation at approximately
1 over 4 t i. The frequency response starts to change drastically compared to what it is at 0, hence if you
look at it this is for the i th path. However, this is for the i th path hence if
you look on at it or all the paths on average that can be set as approximately 1 over 4
sigma tau this is the frequency where the frequency response is significantly different compared to the frequency
response. Here, at 0 this can also be termed as this is can be termed as the frequency
of significant change. Now, you can see you can also say before this point of significant change the frequency
response is approximately flat and it starts falling after this point of significant change. Hence, the coherence bandwidth is nothing
but
twice this frequency of significant change which is twice 1 over 4 sigma tau which is
essentially 1 over 2 sigma tau where sigma tau is the is the RMS delay spread. Look at this, let me just repeat
this argument again closely remember this is only an approximate argument. It is not
an exact argument, I am saying, let us look at the frequency response the frequency response contains components
which are a i e power minus j 2 pi f tau i at f equals 0. This is a i at f equals one
over four tau i this is j a i or minus j a I, so it has change significantly from f equals 0 to f equals
one over four tau i. Hence, the point of significant change of
this spectrum is approximately one over four tau i for the i th component if i look at
the channel on an average the point of significant change because look at this tau i is nothing but the delay of the
i th component. So, the point of significant change can also be set to be 1 over 4 sigma
tau on an average were tau sigma tau is the RMS delay spread. Hence, I can say the coherence band width is twice
this point of significant change the coherence band width is essentially this side plus this
side put together it is twice 1 over 4 sigma tau or 1 over 2 sigma tau. Hence, i can write coherence, remember
these are all only approximate relations, hence I can write coherence bandwidth as approximately
equals to 1 over 2 sigma tau. So, let me again summarize this relationship
here B c equals 1 over 2 sigma tau where B c remember B c. This is the relation for coherence
band width B c equals the coherence bandwidth of the system sigma tau equals the RMS delay spread.
Even though this is an approximate relation this has been found to be true in practice
in 3 G 4 G wireless communication scenario. So, 2 G wireless communication scenarios for a great degree
to a great degree of accuracy and this can be used as a rule of thumb we can also look
at this intuitively. This says essentially that B c the coherence band width is inversely proportional to the delay
spread, let us look at this. In a traditional wired communication channel
we have only a single path that is an impulse hence if I look at this frequency response
let me look at a wireless communication system. This is time this is frequency in time the wired communication
system is an impulse that is its delay spread is 0 its coherence band width is infinity.
The frequency response of an impulse is nothing but a frequency flat spectrum implies its coherence band width
is infinity. Now, as the delay spread starts to increase
that is it starts to spread in time what happens in frequency in frequency it starts to shrink
because an as the time response starts to get wide and remember for the ideal case where the time
is an impulse frequency. It is flat it is infinity or the coherence band width is infinity
as the time response starts to shrink the as the time response starts to expand the frequency response starts to shrink
that will be something like this. So, as the time response increases the coherence band
width decreases in the limit when the time is flat that is when the time spread is infinity when I have sigma
tau equals infinity. It has an infinite spread in time in frequency
it is simply an impulse which means B c equals 0 hence look at this the delay spread and
coherence band width are inversely proportional to each other in time. When I have a single impulse that is
0 time spread then my frequency spread is infinity in the limit, when i have infinite
time spread that is the RMS delay spread is infinity. I have an impulse in frequency which means my spread in frequency
is 0 which means my coherence band width is 0. So, essentially from these figures what
I can see intuitively is B c is inversely proportional to sigma, now let me write that down again over here
clearly. The coherence band width is inversely proportional
to the RMS delay right which is B c the coherence band width is inversely proportional to sigma
tau. We said one approximation that we can use is that B c equals 1 over 2 sigma tau that is
the coherence band width is 1 over twice the RMS delays spread. This is the approximation
that we can use in our computation, now let us look at the last factor in this discussion of coherence band
width and delay spread which is the most important factor. These all come together what is the
relation between this B c and sigma tau we have sigma tau we have looked at it in the frequency domain. Now, let us look at what that relationship
is in the time domain what is or in other words simply relation between B c coherence
band width and sigma tau the RMS delay spread what are its implications what is it is what are the relation
in time domain. What is the relation between these two quantities in the in the time domain,
now let me consider a signal at transmitted wireless signal let me say this signal let me consider a base
station. Let us go back to the earlier example of a
base station and a wireless receiver, let us say I have a direct path and I have several
scatter paths arising after the buildings. So, this is the direct path there are several scatter paths, now let us say
I have a transmitted signal that looks like this. Let us say this is my transmitted signal
this is symbol 0 this is symbol 1, this is symbol 2 this is symbol 3 this is symbol 4, let us say this is my transmitted
first the copy of the signal which is from the directed direct path. So, I get symbol
s 0, s 1, s 2, s 3, s 4 so on. So, this is the one of the copies of the signal which is arriving from the direct path, then I have
another copy of the signal which is arriving at a slight delay look at this signal the second signal which
also contains s 0, s 1, s 2, s 3, s 4. Now, this signal which is corresponds to the
scatter path, so what I am saying, I am saying I have two paths, one is the direct path between
the base station and mobile that is giving me a signal. Then, there is also a signal copy that is
the same signal which is arriving at a slight delay that corresponds to a scatter component
this signal is exactly the same as this signal. This copy is slightly delayed from this copy, there is a slight
delay let me call this t d that is the delay. Now, you can also see that this delay is nothing
but the delay spread of the channel so t d equals sigma tau which is the delay spread of the channel.
So, I am saying I am transmitting the signal at the transmitter at the base station, I
am receiving copies, I am receiving one signal. Then, I am receiving another copy which is
coming from the scatter component that is from trees and buildings,
but it has to travel a larger distance. So, it is arriving at a slight delay that delay
is nothing but the delay spread of the channel. So, now due to lack of time I have to conclude this lecture at this point
we will take up this lecture from the next. In the next lecture we will take up we start
at this point and continue this discussion further about the relation between the RMS delay spread and the coherence
bandwidth what is the intuitive relation between these two points.
Thank you very much.

## 10 comments on “Mod-01 Lec-10 Coherence Bandwidth of the Wireless Channel”

1. bonhomie9 says:

Thank you. Your lectures just saved me from an exam that could have gone awfully bad! The flow of the concepts – one leading to the next was great and easy to follow.

2. Nabil Akdim says:

Thanks a lot for this valuable content, any chance to get the powerpoint slides ?

Thanks again

3. Zahid Lone says:

Simply awesome sir.Thanks alottttttttttttttttt

4. Meme Saad says:

Thanks Mr…. your explanations are awesome.. really many thanks

5. Sylar Lao says:

at 47:02, how exactly do we use RMS delay spread instead of tao i?

6. shivik isrie says:

HOw can I detremine the coherence bandwith from S21verus frequecny (frequencsy sweep) measuremnets using matlab. Since I am new to matlab. I have no idea how to formulate it in matlab?

7. benjahnz says:

You sir are one of my top 3 youtube comms guys. Hats off to you.

8. Dr B Siva Kumar reddy says:

Perfect explanation…. Iam a big fan of u sir…

9. venkat sahan says:

Sir,
At 44:54 you wrote "1/4*tau i" approximately is "1/4*sigma tau". Here "tau i" is a random variable and "sigma tau" is a standard deviation. How can random variable be approximated to standard deviation?

10. MsMammeta says:

Thanks you, it really helped ❤️