Mod-01 Lec-11 ISI and Doppler in Wireless Communications


Welcome to the course on 3G and 4G wireless
communication systems in the last lecture we completed our discussion or we completed
one part of our discussion on the RMS delay spread we said the RMS delay spread is the interval of time
over which the signal copies are received in the wireless communication channel because
there are not just once there is not just one signal component, but there are several signal components corresponding
to the direct path and the scattered paths we said the delay spread is the interval over
time over which these signal components are received we also said in typical outdoor wireless communication
channels wireless communication systems. The delay spread sigma tau is approximately
1 to 3 micro seconds we also said that if I take the Fourier transform. If I look at
the Fourier transform of the delay profile I get the Fourier transform of the delay profile that is H(F) and we defined
the coherence bandwidth as that bandwidth over which this frequency response is approximately
flat we said that is the coherence bandwidth of the wireless channel that is the bandwidth over
which the response of the channel delay profile is approximately flat and more importantly
we said a very important thing. About flat fading and frequency selective
fading we said if the bandwidth of the signal being transmitted is less than the coherence
bandwidth then that system and the channel is a flat fading channel in this case there is no distortion however
if the signal bandwidth is greater than the coherence bandwidth as in this case. When the signal bandwidth is greater than
the coherence bandwidth then there is attenuation at the edges which implies that there is frequency
selective distortion. Hence at the signal bandwidth Bs is greater than the coherence bandwidth there
is frequency selective distortion and lastly. We also said that the coherence bandwidth
and the delay spread are inversely related to each other that is if the coherence bandwidth is high then the delay
spread is low and similarly, if the coherence is low the delay spread is high and more specifically
an approximate relation between coherence bandwidth and delay spread is coherence bandwidth.
Bc is equal to 1 over 2 sigma tau this is employed very frequently and to a high degree
of accuracy this characterizes the relation between the coherence bandwidth and the delay
spread sigma tau in a wireless communication system now with that
let us go on to today’s discussion which is we were starting to explore the relationship
between the time domain between the coherence bandwidth and delay spread what does it mean in. The
time domain we looked at the frequency domain interpretation of this now let us look at
what is happening in the time domain? So, let us look at relation
in time domain and for that we said we will
start by considering a signal which has symbols S0. So, this is a digital communication signal
it has symbol S0 for S1, S2, S3, S4 so, S0 is a first symbol S1 is the second symbol
S2, S3, S4 are subsequent symbols that are being transmitted. Now, this is transmitted
from the transmitter what I receive at the receiver is one copy of this from the direct path between the transmitter
and the receiver and then I will receive another path from the scatter component let us say
I have one direct path and one scattered component then I will receive another path from the
scattered component which is slightly delayed with respect to this signal that will look
something like this it is a same signal except it is slightly attenuated and delayed compare to the original signal.
So, that will look like S0, S1, S2, S3, S4 look at this signal look at the second signal
it is the same as the original signal so, if this is received from the direct or line
of site path this corresponds to the non line of site path or the from the scattering
we said we received multiple components of the transmitted signal one from the direct
path and several from the scatters along the scattered paths and the signals from the scattered components are
delayed with respect to the line of site signal look at this it is exactly the same signal
except it is slightly delayed and this delay is nothing, but the delay spread sigma tau of the system and this interval
of the signal is nothing, but the symbol time of my transmitted digital communication signal
alright now look at this I have a signal and a delayed copy arising from the scatter.
Scattering in the wireless channel at the receiver these signals the direct component
and the scattered component are going to add up and that is going to result in interference
at the receiver, but look at this when I add these 2 symbols up
because the delay is small because sigma tau is small when I add these symbols up when
I add these signals up S0 interferes with S0, S1 interferes with S1, S2 interferes with S2, S2 that is the same
symbol is interfering with each at each time instant because sigma tau is very small so,
sigma tau is very small more specifically sigma tau is much smaller than T symbol where T symbol is the duration
of the digital communication symbol. So, I am saying when this delay spread sigma
tau is less than T symbol where T symbol is the time duration of the digital communication
symbol then the at the receiver because of multi path interference the same symbol is interfering
at every time instant that is S0 is interfering with S0, S1 is interfering with S1, S2 is
interfering with S2, S3 is interfering with S3 and so, on now let us consider a scenario in which the delay spread
is much greater than the symbol time now let us consider a scenario in which the delay
spread is much greater or comparable to the symbol time. Let me write this as greater than symbol time
I have a signal that is received from the direct path that is S0, S1, S2, S3, S4 and
so on. However the second symbol here second signal here is delayed significantly compared to the first one. So,
this is my direct signal this is my scattered signal and look at the delay between these
2 signals the delay between these 2 signals is now sigma tau which is much larger than the symbol duration here
I will have S(-1) so, I am saying I have a signal that I am receiving from the direct
path or line of side and I have a delayed signal that is received from the non line of site or the scatters or the scattering
in my 3G, 4G wireless system. Now, when this signals interfere at the receiver
because this delay sigma tau is greater than the symbol time look at this S0 will add with
S minus 1, S1 will interfere with S0, S2 will interfere with S1, S3 will interfere with S2. So, what is happening
is the previous symbol is interfering with the current symbol look at this in this case
what I had was that S0 is interfering with S0, but in the previous case S0 was interfering with S0, S1 was interfering
with S1, S2 was interfering with S2. However, in this case a 0 is interfering with S minus
1, S1 is interfering with S0, S2 is interfering with S1 that is the previous symbol is interfering with the
current symbol hence this is resulting in what is known as inter symbol interference.
Hence, when the delay spread is much smaller than the symbol time there is no problem all
the sigma there is no inter symbol interference, but when the delay spread becomes larger than
the symbol time what happens is the past symbol
interference interferes with current symbol in fact if you have more signal copies and
the delay spread is very large then you have the second path symbol that is S(-2) interfering with S(0)
and so, on. So, as the delay spread increases what happens is we will start having progressively
worser and worser inter symbol interference at the receiver. So, let me write this down clearly in this
slide over here sigma tau that is the delay spread this is the delay spread is greater
than or let us say is much greater than the symbol time this is the symbol time this leads to nothing, but this leads
to ISI or in other words inter symbol interference so, the delay spread much larger than the
symbol time leads to inter symbol interference. Now let us take this idea little bit further
sigma tau greater than T symbol that is the symbol time leads to inter symbol interference
that is what we have to start now let us take the reciprocal of this if I bring T symbol to the left hand side sigma
tau to the right hand side I get 1/T symbol is greater than one by sigma tau which implies
leads to inter symbol interference, but now look at this quantity one over T symbol what is this quantity one
over T symbol this is nothing, but the bandwidth of the signal if I have a signal with symbol
time T symbol the bandwidth of the signal required is nothing, but B signal B signal.
So, one over to symbol is nothing, but the bandwidth of this signal so, from your digital
communication system theory you know that if I am trans I have bandwidth Bs the symbol
time that is possible is one over Bs and look at one over
sigma tau what is one over sigma tau remember we said Bc equals one over 2 sigma tau hence
one over sigma tau is nothing but, twice Bc hence if Bs greater than twice Bc that has inter symbol
interference, but look at this condition what is this condition Bs is greater than twice
Bc. Remember, we looked at this condition before
Bs is greater than twice Bc implies Bs is greater than Bc that is nothing, but the condition
for frequency selective distortion so, Bs greater than Bc implies one over T. Sym greater than sigma one over
sigma tau implies sigma tau greater than T sym implies ISI so, if I follow the argument
from here to hear what I get essentially is let me write that down. Now frequency selective distortion
frequency selective distortion in frequency
domain implies inter symbol inter and this is the surprising conclusion that we can derive
which is essentially that if I have frequency selective distortion in the
frequency domain. That is if Bs is greater than Bc in the frequency
domain I will have frequency selective distortion what which means in the time domain that results
in inter symbol interference. Now similarly, if I have Bs less than Bc then in frequency domain it
is a flat fading channel in time domain it means there is no ISI or no inter symbol interference.
So, that is beautiful conclusion or the very intuitive result that we can derive that is frequency selective
fading and inter symbol interference are inherently related to each other which means, if the
signal bandwidth is greater than the coherence bandwidth.
What I will have is frequency selective distortion which essentially means inter symbol interference
also if Bs is less than Bc that is the signal bandwidth is less than the coherence bandwidth
then the channel is a flat fading there is no distortion
in frequency in time also there is no ISI no inter symbol interference which means no
distortion in time also, these are 2 intuitively or inherently related ideas and its essentially very important in
a 3G, 4G wireless communication system in any wireless communication system to understand
that, there is a very inherent and intuitive relationship between frequency selective fading in the
frequency domain and inter symbol interference in the time domain both of these are essentially
the time and frequency domain analogues of each other or counterparts of each other so, it is important
to understand the relation between them. Now let us do an example to sort of solidify
our understanding of coherence bandwidth and frequency selective fading so, let us do an
example
we said outdoor channel in and outdoor channel the delay spread typical delay spread sigma tau
is approximately of the order of micro seconds that is for outdoor channels the delay spread
is approximately one micro second now let us compute the coherence bandwidth corresponding to this
delay spread alright we know that the coherence bandwidth Bc is one over 2 sigma tau which
is one over 2 times one micro second this is nothing but, one over 2 micro seconds which is essentially
500 kilo hertz so, I am saying the coherence bandwidth in and outdoor 3G, 4G wireless communication
channel is approximately 500 kilo hertz so, the coherence bandwidth is approximately 500 kilo
hertz now let us compare it with our known communication systems. Let us look at GSM has a bandwidth of 200
kilo hertz roughly 200 kilo hertz which implies Bs equals 200 kilo hertz now remember coherence
bandwidth is 500 kilo hertz so, Bs is less than Bc look at this Bs is 200 kilo hertz Bc is 500 kilo hertz
Bs is less than Bc hence a GSM or a 2G wireless communication system remember this is a 2G
wireless communication system this is a flat fading flat fading system or this is also a system in which there
is no ISI in GSM because its bandwidth 200 kilo hertz is much smaller than the coherence
bandwidth which is 500 kilo hertz hence this 2G wireless system is flat fading or there is no inter
symbol interference now let us look at a 3G wireless system. Let us look at a WCDMA system remember WCDMA
is a 3G wireless system in fact it is a spread spectrum system, a wide band spread spectrum.
It is the spread spectrum system which means its bandwidth is huge its bandwidth is about its
bandwidth is 5 mega hertz, which is also can be written as 5000 kilo hertz alright its
bandwidth is 5 mega hertz each mega hertz is 1000 kilos. So, its bandwidth is 5000 kilo hertz this is BSWCDMA
now look at this is 5000 kilo hertz much greater than 5000 kilo hertz which is the coherence
bandwidth. So, Bs for WCDMA is much greater than Bc so,
Bs is 5 mega hertz which is much greater than Bc which is 500 kilo hertz hence by definition
by our definition this results in frequency selective fading in the frequency domain in the time domain there
is going to be ISI or inter symbol interference and this is a 3G system. So, the difference
here essentially compared to a 2G system is because the bandwidth is higher this bandwidth is much
greater than the coherence bandwidth hence resulting in frequency selective distortion
and also ISI which is bad in wireless communication system because.
Now, we have to employ some technique at the receiver so, that we can reverse this distortion
in the frequency domain. This technique is employed to reverse this distortion is known
as an equalizer, because the distortion is frequency selective
we have to equalize the frequency response at the receiver that is known as an equalizer.
However, we will not go in detail into the discussion of that equalizer, because that is over that is relevant
to the or you should have studied that in your course on digital communication system,
but here we will just note that when the distortion is frequency selective.
When there is inter symbol interference we will need an equalizer or equalization process
at the receiver to undo this distortion alright so, with that we complete our discussion on
the delay spread. I want to again point out that the delay spread
in summary delay spread is a important parameter in a 3G, 4G wireless communication system
because it characterizes that interval of time over which you are receiving copies starting from the
direct component to the scatter components, and the delay spread is inherently related
to the coherence bandwidth as this delay increases the coherence bandwidth decreases and if the signal bandwidth
is less than the coherence bandwidth that is fine it does not result in frequency distortion.
But if the signal bandwidth is greater than the coherence bandwidth then we have a problem
that results in frequency selective distortion and its counterpart in time is essentially
inter symbol interference if the signal bandwidth is greater
than the coherence bandwidth it results in inter symbol interference in time domain which
is bad because symbols start interfering with one another making detection impossible or detection erroneous
at the receiver now let us go to the other aspect of a wireless communication system. Which is also important which is the DOPPLER
shift so, this DOPPLER are essentially the DOPPLER shift, we know from a basic discussion
of high school physics that DOPPLER shift is nothing, but the apparent or the relative change into the change
in frequency. Because of relative motion between the transmitter and the receiver so, here
let me draw a base station in a wireless communication system which is acting which is transmitting
a signal. Let me draw a signal a base station is transmitting
a signal and there is a receiver which is moving with a velocity towards the base station.
so, this is a base station which is transmitting a signal. This is a mobile station which is moving towards
the base station or it can also be moving away from the base station, but the motion
has to be relative that is this is the transmitter this is the receiver there is relative motion between these that
is one is moving towards or away from the other. In fact it can be a scenario in which
the base station is also moving that is a complicated scenario it does not occur in a wireless cellular network.
But it can pretty much occur for instance in a wireless sensor network or a mobile wireless
adhoc network in such scenarios so, when there is relative motion between the transmitter
and the receiver this results in a change in the frequency
that is received at the receiver that is known as the DOPPLER shift. So, that let me write it down formally the
DOPPLER shift can or the DOPPLER shift is the change in the frequency of the electromagnetic
wave that is arising
due to relative the key word here is relative motion between the transmitter and
receiver. So, DOPPLER shift is nothing, but the change in the frequency of the electromagnetic
wave as we said it is the change in the frequency that is arising due to relative motion between the
transmitter and the receiver and if for instance. Let me draw transmitter if my transmitter
is here and the receiver is here and the receiver is moving towards the transmitter then the
received frequency is higher and if the receiver is moving away from the transmitter then the received frequency
is lower. So if the motion is towards the transmitter then the received frequency is
higher if the motion is away from the transmitter. Then the received frequency is lower now let us go
into how to let us look at how to compute the DOPPLER frequency shifts. So, now let us look at computation
of the DOPPLER
let us look how to compute the DOPPLER shift
let me consider a base station will be simply consider a transmitter which is transmitting
to a mobile station. A base station which is transmitting
to mobile station and this mobile station is moving at an angle of theta.
So, this is my base station let me mark it clearly this is my base station this is my
mobile station or my mobile terminal this is moving such that if I look at the direction
of the velocity and if I look at the line joining the base station and the
mobile station the angle between the velocity and the line joining the base station and
mobile station is theta. So, this angle here is theta now the DOPPLER
shift Fd so, the DOPPLER shift Fd in frequency Fd is given as Fd equals v cosine theta over
c divided by v cosine theta divided by c into Fc. Where c is v is the velocity theta is the angle c is the speed
of light or the speed of electromagnetic wave in free space. We know that and Fc is the
carrier frequency. So, let me write down this v is already indicated here that is the velocity of the mobile terminal
theta is the angle between the velocity and line joining the mobile station and the base
station c equals speed of light
and Fc equals the carrier frequency. And then the received frequency at the mobile
is given as F the received frequency is given as F received is Fc the carrier frequency
plus the shift the DOPPLER shift which is Fd which is equal to Fc plus vcos theta over c times Fc. Now you can
see from this if 0 less than theta less than pi by 2 the mobile 0 less than theta less
than pi by 2 then the mobile is moving towards the base station, which means cosine theta if 0 less than theta less
than pi by 2. MS is moving towards the BS and hence cosine theta is positive because
0 less than theta less than pi by 2. Cosine theta is positive.
Hence the frequency shift is positive or the frequency is higher. Similarly, if pi by 2
less than equal to theta less equal to pi then the mobile is moving away from the base
station. Cosine theta is negative for theta pi by 2 less than equal
to theta less than pi. Hence Fc minus something hence the received frequency low. So, when
the mobile is moving away from the base station the received frequency is lower that can be represented
succinctly by this formula. In fact you can see one thing very interesting
if theta equals pi by 2. Look at this if theta equals pi by 2 then cosine theta is 0 which
means if the motion here is such that it is perpendicular to line joining the base station and the mobile then
cosine theta is 0. Hence even though there is a velocity there is no effect on the received
frequency is exactly the same. So, this say something very interesting which
theta equal to pi by 2 even though there is a positive velocity the received frequency
is exactly the carrier frequency. So, as a function of theta this is the received frequency at the mobile due to the
DOPPLER shift. This is received frequency due to DOPPLER. Now let us do an example, let us consider
a carrier frequency Fc equals 1850 mega hertz. And let us consider a vehicle moving at 60
miles per hour directly towards the base station which implies theta equals 0. So, what am I considering
I am considering Fc equals carrier frequency 1850 mega hertz, a vehicle which as a person
in it whose has a mobile station who has the mobile station who is moving at 60 miles per hour directly towards
the base station. We want to compute what is the DOPPLER shift? So, we want to compute
DOPPLER shift and the received frequency and we know that the DOPPLER shift FT. So, first let us look at this velocity is
60 miles per hour. Let me convert this into standard meters per second. Each mile is roughly
1.61 kilometer. So, this is 60 into 1.61 kilometers per hour which is equal to 60 into 1.61 into 5 divided by 18
meters per second which is equal to 26.8 meters per second. So, 60 miles per hour is 26.8
meters per seconds. Let me write that down 60 miles per hour equals 26.8 meters per second
which means. Now Fd equals v by c cosine theta into Fc,
but theta equal 0. So, cosine theta equal 1. So, this is simply 26.8 that is the velocity
divided by 3 into 10 power 8 that is the velocity of light into Fc which is 1850 mega hertz. Remember we said the carrier
frequency is 1850 mega hertz. So, this is simply 1850 into 10 power 6 this is nothing,
but 165 hertz. So, the DOPPLER shift Fd equals 165 hertz. And the received DOPPLER frequency F received
is Fc plus Fd which is 1850 mega hertz plus 165 hertz.
So, the DOPPLER shift we have calculated is 165 hertz and the received frequency because
remember theta is 0 the mobile is moving directly towards the base station. So, the shift is
positive so, the received frequency is higher it is 1850 mega
hertz plus 165 hertz. So, the DOPPLER shift is 165 mega hertz. I compared to the carrier
frequency which is 18.1850 the DOPPLER shift is 165 hertz compared to the carrier frequency which is
1850 mega hertz it might seem that this DOPPLER shift is very small compare to the frequency
of the carrier, but we are going to see later that this small shift itself has a significant impact on the
3g 4g wireless communication system. If you are going to look at this a either
this lecture or subsequent lecture for this small shift will have a significant impact
on the communication the wireless communication system. Now let us go towards developing a model try to understand
this nature of the DOPPLER shift bit more in depth. So, let us start go back to our
base band. So, let us go back to our base band channel
and we said that each path is characterized by ai delta t minus tau i. Where ai is the
attenuation of the ith path and tau i is the delay of
the ith path. Now consider a scenario in which there is mobile
and it is moving directly towards the base station. So, there is a mobile it is moving
directly towards the base station. Now let us say initially the delay of this
is tau i this path is the ith path this as a delay tau i. Now after t lets say this is
moving with a velocity v after t the distance decreases by v t. So, after t distance decreases by vt. Hence when the distance decreases
the delay also decreases. Hence the delay of propagation on this path decreases by tau
i minus vt divided by c. So, as the delay is as the distance is decree as the distance is decreasing the
delay is also decreasing. Now let us look at another case where this
is the mobile station this is the base station and the mobile is moving at an angle of theta.
The component of velocity in that line joining the mobile station and base station is v cosine theta. So, the
velocity in this direction is v cosine theta. Hence in a time T the distance decreases the
delay decreases as tau i minus v cos theta into t that is the decrease in the distance between the mobile station and
the base station divided by c. So, this is now the new delay let me write
this tau i as a function of time. So, what am I saying the delay initial delay this is
tau i which is the initial delay. However this is not constant because the mobile itself is moving at with a velocity
v which an angle theta. Hence this delay is now changing with respect to time. How is
this delay changing with respect to time tau it the delay at time t is tau i minus v cos theta t over c. Where v is the
velocity theta is the angle of the velocity with the line joining the mobile and base
station. So, this is interesting now we are saying
that that the delay of this channel is not constant, but on the other hand it is a function
of time. Now let us look at our earlier expression for the flat fading channel coefficient. So, remember the flat fading channel coefficient
h which is also the flat fading channel coefficient remember this is given as i equals 0 to l
minus 1 aie power minus J2 pi Fc tau I remember the flat fading coefficient h is given as ai, where
ai is the attenuation of direct path e power minus J2 pi Fc tau i, where tau i is the delay
of the i th path sum from i equal 0 l minus 1.
Now something interesting happens because now this delay itself is a function of t which
means I have to write this as tau i of t. Now let us simplify this expression further
in the next page. So, the flat fading channel coefficient now
when the mobile is moving remembered with respect to the base station in the 3g 4g wireless
communication systems becomes i equal 0 to l minus one aie power minus J2 pi Fc remember tau i is now
tau i t which is tau i minus v cosine theta over c times t. This is now equal to i equals
0 to l minus 1 aie power minus J2 pi Fc tau i into e J2 pi Fc v cos theta over c times t. However realize that Fc into
v cos theta over c we have seen this before this is simply the DOPPLER shift Fd.
So, Fc into v cos theta over c, v cos theta over c is Fd hence this can be written as
i equals 0 to l minus 1 aie power minus J2 pi Fc tau i into e power J2 pi Fdt, where
Fd is the DOPPLER shift. Now look at this earlier this was a constant with respect
to time now there is a factor which is time varying so, this is a time varying phase.
So, the phase is varying with time which means the phase of every component in this flat
fading channel is varying by time because of the motion between the mobile station and
the base station which means the whole channel is now varying
with time which means this is now not simply h, but this is h of t which is a time varying
channel. So, what has DOPPLER resulted in DOPPLER which is caused by the motion between the mobile station
and base station has essentially resulted in a time varying channel. Hence let me summarize
that observation over here. The DOPPLER shifts or DOPPLER implies time
varying in fact we said earlier that the channel is a constant, but now however because of
DOPPLER because of the relative motion between the mobile station and the base station. My channel has
acquired a time varying character. So, the DOPPLER frequency shift is essentially resulting
in the time varying nature of the channel thus mobility results in DOPPLER. So, mobility results in DOPPLER
which in turn results in a time varying. So, mobility results in DOPPLER which in turn
results in a time varying channel. And this also known as time selectivity time
varying channel also known as technically as a time selective channel remember if a
channel is varying in time. It looks something like this is the channel
coefficient it is varying with time. So, as time it is varying which means it is a different
time it has different value. So, it is selective remember we saw earlier frequency selective that is if it is varying
in frequency that is a one frequency it is flat in the coherence bandwidth, but outside
that it is varying in frequency. Hence it is selective in frequency now we are saying in the case of DOPPLER which is
it is there is something that happens similar in time which is it is varying in time.
So, it is selective in time so, a time varying channel also known as time selective. Let
me summarize that over here this implies essentially a time varying essentially implies time selectivity.
So, DOPPLER variation in time of DOPPLER in time
results in a time varying or a time selective channel. Now let us consider another let us
consider what is happening to each coefficient. Let us look at what is the relation at each
coefficient let us look at ai of t that is given as aie to the power of minus J2pi Fc
tau i into e power J2pi Fd of t. So, this was the flat fading coefficient earlier. Now it is varying with respect to time. The
varying factor is e power J2pi Ftd. Now let us look at t equals 0 at t equals 0 this is
aie power minus J 2pi Fc tau i times e power J2pi Fdt which is e power J2 pi F d0 which is 1.
Hence at t equals 0 ai of 0 is simply equal to aie to the power of minus J2 pi Fc tau
i. Now let us look at what happens a t equals 1 over 4 four Fd ai of 1 over 4Fd equals aie
to the power of minus J2 pi Fc tau i times e to the power of J2 pi Fd
1 over 4 Fd which is equal to aie power minus J2 pi Fc tau i into 2 pi Fd into1 over 4 Fd
is pi over 2. So, this is eJ power pi over 2 which is nothing, but J aie power minus J2 pi Fc tau i.
Now look at these 2 values if you look at these 2 values at t equals 0 it is aie power
minus J2 pi Fc tau i at t equals one over 4 d it has become J aie power minus J2 pi
Fc tau i. So, it has changed drastically for instance if ai is real at
this point it has changed to J i i ai which is an imaginary number. So, the phase and
in fact the quantity has changed drastically in one over 4 d. So, similar to what we did in the context
of coherence bandwidth we can say in t equals 1 over Fd in time interval equals 1 over Fd.
My ai of t is changing is changing drastically or in other words the channel can be assumed to be constant in one
interval of t equals one over 4 d and then it is changing in the next interval of one
over 4 Fd and so on. and so forth. So, we can say it as the channel is changing after every one over 4 d drastically
and it is constant over every interval approximately constant over every interval of 1 over 4 F
d and this is known as the coherence time. So, coherence time equals Tc equals 1 over
4Fd which means the channel is approximately constant in this interval of length one over
4 Fd that is 0 to 1 over 4 F d in the next interval of 1 over 4 F d to twice 1 over 4 F d that is over 2 F d it is
changing then 1over 2 Fd to 3 over 4 Fd it is again changing and so on.. And so forth if I look at the channel and
if I take time in 0 to 1 over 4 Fd it is approximately constant in the next interval one over 4 Fd
to 1 over 4 2fd which is of duration 1 over Fd it is again approximately constant and again in another
interval of duration 1 over 4 Fd it is approximately constant and it is changing from interval
to interval this interval of duration 1over 4 Fd is known as Tc equals the coherence time remember coherence
bandwidth is that bandwidth over which the frequency response is approximately constant.
Coherence time is the time over which the channel in time is approximately constant. Tc equals coherence time equals time over
which channel is approximately constant let Tc is the coherence time over which the channel
is approximately constant. So, at this point let us end todays lecture and we will take up this discussion
about coherence time detail again starting with the next lecture.
Thank you.

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