WEBVTT
Kind: captions
Language: en
00:00:19.400 --> 00:00:36.489
ok so we begin this course which is given
the name discrete time signal processing which
00:00:36.489 --> 00:00:42.740
is a discrete time it means the signal normally
we come across analog signals analog signals
00:00:42.740 --> 00:00:48.650
are actually wave forms wave forms of time
in general time it could be space also like
00:00:48.650 --> 00:00:54.720
if it is an image continuous image it is a
function of two xs x and y is a continuous
00:00:54.720 --> 00:00:59.500
function that's an analog image similarly
an analog wave form as a function of time
00:00:59.500 --> 00:01:07.360
could be like this there are analog wave form
you can call it x subscript a for analog as
00:01:07.360 --> 00:01:14.790
a function of time ok so the function of time
t is a continuous variable time t are you
00:01:14.790 --> 00:01:19.549
plot it is called a analog signal it can come
from say somebodys speech like i am saying
00:01:19.549 --> 00:01:23.440
something if you would record the wave form
as a function of time it will be like this
00:01:23.440 --> 00:01:29.970
it can get from any source from communication
source from you know from various sources
00:01:29.970 --> 00:01:35.300
it just convert it into an wave form function
of time so for us signal means analog signal
00:01:35.300 --> 00:01:40.590
means something which is a function of time
of function of one variable either normally
00:01:40.590 --> 00:01:46.240
we deal with functions of time or in the case
of images function of space but they are at
00:01:46.240 --> 00:01:53.790
least function of two variables x axis y axis
something like if x comma y x is continuous
00:01:53.790 --> 00:02:01.920
in x y is continuous at each x y there is
a value pixel value i mean intensity value
00:02:01.920 --> 00:02:06.520
and since x and y are continuous there will
be a continuous image there is a two dimensional
00:02:06.520 --> 00:02:12.840
signal one dimensional signal are typically
function of time ok analog these are analog
00:02:12.840 --> 00:02:18.849
but if i discretize the time that is i will
not observe it at all points i will observe
00:02:18.849 --> 00:02:26.470
it at uniformly separated points may be zero
then t then two t what capital t is called
00:02:26.470 --> 00:02:31.310
the period sampling period that i am taking
a sample here next sample here next sample
00:02:31.310 --> 00:02:47.290
here next sample here dot dot dot dot so the
sample values when i consider the sample values
00:02:47.290 --> 00:02:53.879
this axis is no longer time we in fact throw
away the notion of time from here we only
00:02:53.879 --> 00:02:59.739
bother about index this is a zeroth sample
so this is n equal to zero this is the first
00:02:59.739 --> 00:03:04.670
sample n equal to one so this gap has nothing
to do with capital p you can make it very
00:03:04.670 --> 00:03:11.109
narrow very wide it gives nothing its only
zero one two three what matters is for this
00:03:11.109 --> 00:03:16.370
first sample which is second sample which
is third sample that is zeroth sample first
00:03:16.370 --> 00:03:21.569
sample second sample third sample like that
so that means this x axis is discrete and
00:03:21.569 --> 00:03:25.930
that is why it is discrete time that is if
the original signal came from a function of
00:03:25.930 --> 00:03:31.849
continuous time then if you discretize the
axis you just get points zero one two three
00:03:31.849 --> 00:03:36.629
ok and corresponding sample where as you plot
you will get a sequence this sequence is a
00:03:36.629 --> 00:03:41.930
discrete time sequence ok we just call it
a sequence and i repeat again there is no
00:03:41.930 --> 00:03:47.249
notion of time here this a n has got no time
its just an integer diamonds are less unit
00:03:47.249 --> 00:03:53.029
less does n equal to zero n equal to one n
equal to two n equal to three so who is zeroth
00:03:53.029 --> 00:03:57.709
who is first who is before whom who is after
whom these are the things matters and then
00:03:57.709 --> 00:04:04.439
i will call it x n this will be a sequence
we also call it discrete time signal
00:04:04.439 --> 00:04:10.599
now advantage of this thing is this representation
is that if you throw either notion of time
00:04:10.599 --> 00:04:17.510
you can take care of any sequence of numbers
because may be a sequence is obtained just
00:04:17.510 --> 00:04:23.550
manually not by certainly by sampling an analog
signal the sequence of numbers them also i
00:04:23.550 --> 00:04:30.310
can process here like i mean if you get a
sequence of numbers one two nine five minus
00:04:30.310 --> 00:04:34.670
seven dot dot dot you automatically generate
or may be your computer is generating sequence
00:04:34.670 --> 00:04:42.240
of numbers i just plot them one after another
zeroth first second i put a zero value i mean
00:04:42.240 --> 00:04:48.110
i set one origin index origin for one of them
zero and then one two three and put them accordingly
00:04:48.110 --> 00:04:53.940
ok so what i get is a sequence so but this
time it is not obtained by sampling any analog
00:04:53.940 --> 00:04:59.400
signal so this is more general if i throw
a either notion of time just begin index n
00:04:59.400 --> 00:05:04.190
and call it zero or one or two or three i
can take care of not only sequences obtained
00:05:04.190 --> 00:05:09.979
by sampling an analog signal but any sequence
sequence manually generated sequence of numbers
00:05:09.979 --> 00:05:19.479
sequence generated by computer ok data generated
by computer all that is why here after discretizing
00:05:19.479 --> 00:05:25.900
we do not write in terms of time we just write
in terms of these now the question is why
00:05:25.900 --> 00:05:31.569
should i suppose i am considering only this
analog signal and a sequence obtained by sampling
00:05:31.569 --> 00:05:33.250
this like this as i discussed
00:05:33.250 --> 00:05:40.710
suppose now why should we go for this discrete
time signal processing that is processing
00:05:40.710 --> 00:05:48.020
this rather than the analog wave form answer
is analog wave form processing beams actually
00:05:48.020 --> 00:05:53.639
what is processing processing means i should
be able to do various operations as many operations
00:05:53.639 --> 00:06:00.180
as possible on that input signal now suppose
these are analog signal i want to multiply
00:06:00.180 --> 00:06:05.570
i want to divide i mean by some number some
fraction this is possible in analog domain
00:06:05.570 --> 00:06:11.159
just take a registering divider r one r two
ok set the values of r one r two so that and
00:06:11.159 --> 00:06:17.229
we get a fraction of that is very easily durable
if you want to multiply by a number this signal
00:06:17.229 --> 00:06:22.060
in my processing involves multiplying a number
greater than one that means amplification
00:06:22.060 --> 00:06:26.240
that is also very easy we pass it through
a common emitter amplifier or common based
00:06:26.240 --> 00:06:31.870
amplifier or amplifier whatever adjust again
so you can do this amplification k so this
00:06:31.870 --> 00:06:36.240
operation where you have to multiply this
by a number greater than one in analog domain
00:06:36.240 --> 00:06:39.629
we can do that no problem using amplifiers
00:06:39.629 --> 00:06:48.180
then suppose you want to add two such wave
forms ok x one x two two such wave forms you
00:06:48.180 --> 00:06:55.979
want to add that is also possible because
what you do there you use an based ok and
00:06:55.979 --> 00:06:59.860
thus give the two signals you get an output
if it is inverted inside you pass it through
00:06:59.860 --> 00:07:04.689
another inverting amplifier and all that so
adder is very must there so i can add i can
00:07:04.689 --> 00:07:09.449
carry out addition operation and thereby i
can carry out subtraction operation also not
00:07:09.449 --> 00:07:15.669
a problem multiplying two signals in analog
domain that is also possible there are multipliers
00:07:15.669 --> 00:07:21.130
and by analog multipliers which can take two
signals fourth quadrant multipliers and get
00:07:21.130 --> 00:07:28.370
you out connecting log of a signal that every
value of this wave form when i take the log
00:07:28.370 --> 00:07:34.629
of it can i do a logarithmic calculation yes
there is something called log amplifier using
00:07:34.629 --> 00:07:40.169
so you can carry out log you can carry out
anti log so these all these are present and
00:07:40.169 --> 00:07:46.689
some more operations are possible one point
is number of such operations is very limited
00:07:46.689 --> 00:07:54.979
you cannot go beyond a few for instance if
you have to take a say tenth root of these
00:07:54.979 --> 00:08:01.490
or even square root cube root can i do that
not easily any other operation if i take this
00:08:01.490 --> 00:08:06.050
value if i have to carry out trigonometric
operation so every value i have to take it
00:08:06.050 --> 00:08:13.430
sin or cos how much this is very difficult
in analog domain you cannot do but if i convert
00:08:13.430 --> 00:08:20.060
them into a sequence so just sequence of pure
numbers and give these numbers to a computer
00:08:20.060 --> 00:08:27.939
then in a computer i can run an algorithm
on them on the sequence and by the in my algorithm
00:08:27.939 --> 00:08:33.180
through this algorithm i can do any kind of
operation all operations are possible so that
00:08:33.180 --> 00:08:38.880
is digital processing processing this sampled
values but moment i make them available in
00:08:38.880 --> 00:08:43.219
this form and give it to a computer computer
can work only on numbers discrete elements
00:08:43.219 --> 00:08:48.380
computer can work on continuous elements so
discrete numbers one after another so my algorithm
00:08:48.380 --> 00:08:53.510
will work out then and give you any operation
you want so the range of operation becomes
00:08:53.510 --> 00:09:00.961
inmates anything is possible if you go for
digital processing but only thing is you can
00:09:00.961 --> 00:09:06.150
ask me a question that look here i am not
taking the entire wave form ok i am not taking
00:09:06.150 --> 00:09:10.350
the entire plot i am probably taking sum of
some values one value another value another
00:09:10.350 --> 00:09:17.490
value another value i am leaving leaving out
this intermediate wave form part intermediate
00:09:17.490 --> 00:09:23.300
values then am i not losing something am i
losing some [informative/important] important
00:09:23.300 --> 00:09:28.910
information because i want who are crazy to
get into some kind of digital processing answer
00:09:28.910 --> 00:09:37.290
is no this is the beauty of digital signal
processing that if suppose this analog function
00:09:37.290 --> 00:09:42.930
has got a finite bandwidth then it can be
shown this is the fundamental theory i mean
00:09:42.930 --> 00:09:49.930
d s p verifies take the samples close enough
that is make the sampling period less than
00:09:49.930 --> 00:09:56.590
equal to some value or sampling frequency
greater than equal to some value some ok there
00:09:56.590 --> 00:10:02.210
the samples are good enough to carry all the
information of the envelope even if the intermediate
00:10:02.210 --> 00:10:10.090
part is that actually what is called nyquist
sampling rate which i will cover later that
00:10:10.090 --> 00:10:14.640
is there is a fundamental limit for band limited
signals that limit is called nyquist rate
00:10:14.640 --> 00:10:21.830
if you sample this analog signal at a frequency
higher than nyquist frequency that means sampling
00:10:21.830 --> 00:10:27.720
period is shorter than that more dates more
close then those samples carry all the information
00:10:27.720 --> 00:10:35.000
of the analog envelope you don't lose anything
that is fundamental that gives you a permit
00:10:35.000 --> 00:10:41.630
a license to replace this analog wave form
by a series of discrete samples and which
00:10:41.630 --> 00:10:47.970
can be easily processed by a computer or by
a dedicated digital hardware and any operation
00:10:47.970 --> 00:10:48.970
can be done
00:10:48.970 --> 00:10:53.550
as you understand in digital processing number
of operations can goes to infinity you can
00:10:53.550 --> 00:10:59.030
do it for anything you want because it will
be basically an algorithm ok algorithm means
00:10:59.030 --> 00:11:04.930
like in computer where you can do anything
the same thing you can work out that but algorithm
00:11:04.930 --> 00:11:09.392
can depend work on continuous stuff that is
why discrete has the big advantage this is
00:11:09.392 --> 00:11:14.900
the main motivation for going into discrete
time signal processing or rather digital signal
00:11:14.900 --> 00:11:21.760
discrete time signal processing right now
there is one more reason while you will go
00:11:21.760 --> 00:11:27.430
for this but before that i am coming to another
point we will coming across two phrases one
00:11:27.430 --> 00:11:32.670
is discrete time signal processing and sometimes
digital signal processing question is are
00:11:32.670 --> 00:11:39.970
these two same actually here in this wave
form this axis is discrete zero one two three
00:11:39.970 --> 00:11:47.490
but this amplitude axis y axis it is not discrete
the sample can take any value but why do you
00:11:47.490 --> 00:11:53.500
give it to computer every sample we pass it
through a quantizer this analog to digital
00:11:53.500 --> 00:11:58.830
converter and then quantizing to a either
sixteen b you know i mean that will give an
00:11:58.830 --> 00:12:03.360
sixteen bit output of the or twenty four bit
output or twelve bit output or thirty two
00:12:03.360 --> 00:12:09.000
bit output so basically this can take some
quantized values only so that time this x
00:12:09.000 --> 00:12:16.800
is also get discretized that is if the y axis
is this anything falling in this range in
00:12:16.800 --> 00:12:21.270
this range may be quantized to will be taken
to be zero value anything falling in this
00:12:21.270 --> 00:12:26.920
range may be take into these value anything
falling in this will taken to these value
00:12:26.920 --> 00:12:33.770
ok and then they are by number of domain some
binary so that is quantization you are familiar
00:12:33.770 --> 00:12:39.430
with the additional quantization so that time
y axis is also discrete so the samples actually
00:12:39.430 --> 00:12:41.020
take discrete values
00:12:41.020 --> 00:12:47.620
so in pure digital processing this x is also
discretized because they take discrete values
00:12:47.620 --> 00:12:51.030
ok because after all they are discretized
by fixed numbers so eight bit numbers or two
00:12:51.030 --> 00:12:55.870
to the power eight sixty four possibilities
only sp sixty four level two will assign ok
00:12:55.870 --> 00:12:59.630
around the level there will be a band anything
falling within the band will be equated to
00:12:59.630 --> 00:13:03.940
that level that another level around that
there is a band anything falling within this
00:13:03.940 --> 00:13:10.350
band will be equated to this level so on and
so forth ok that is why this x is also discretized
00:13:10.350 --> 00:13:17.180
this will be a pure digital signal where both
x axis that is this axis of n at the y axis
00:13:17.180 --> 00:13:24.030
they are discretized but the problem is if
you really start using y axis to be discretized
00:13:24.030 --> 00:13:31.450
also x axis to be discretized also then analysis
other things become very difficult that is
00:13:31.450 --> 00:13:38.690
why we assume that number of bits after quantization
number of bits is very large so therefore
00:13:38.690 --> 00:13:46.250
this bands are very narrow ok are are every
the levels are very close to each other and
00:13:46.250 --> 00:13:50.920
bands that is instead of being two levels
being here they can be very close to each
00:13:50.920 --> 00:13:57.060
other almost adjacent to each other and therefore
they are almost like continuous that is if
00:13:57.060 --> 00:14:01.510
you assume that number of bits is large the
gap between two levels discrete level will
00:14:01.510 --> 00:14:07.700
be very small so you can assume the axis to
be almost continuous and then this becomes
00:14:07.700 --> 00:14:11.470
just a discrete time signal continuous amplitude
but discrete time signal
00:14:11.470 --> 00:14:16.130
we have made this assumption in doing all
analysis and design and other things because
00:14:16.130 --> 00:14:21.060
if you want to carry both the discrete in
time discrete in amplitude then it becomes
00:14:21.060 --> 00:14:26.740
difficult but there again the question comes
that i am making an assumption so there is
00:14:26.740 --> 00:14:31.130
an approximation after all y axis is also
discretized but i am making it you know i
00:14:31.130 --> 00:14:34.720
am assuming it to be continuous under the
assumption number of bit is large but have
00:14:34.720 --> 00:14:42.010
i not introduced them either as so after all
making this assumption that y axis is continuous
00:14:42.010 --> 00:14:49.290
and then going with my design and other things
then we get take a loop and try to estimate
00:14:49.290 --> 00:14:55.050
how much error in my design has got it because
of this assumption that y axis actually is
00:14:55.050 --> 00:15:00.590
discrete but i have assumed to be continuous
ok because the levels are very close to each
00:15:00.590 --> 00:15:06.740
other so that analysis is called finite analysis
or you know finite analysis that we estimate
00:15:06.740 --> 00:15:14.300
ok because of this is the approximation how
much error going ok so again by that we estimate
00:15:14.300 --> 00:15:18.930
with our that error that is ok or not but
before that till design level we assumed it
00:15:18.930 --> 00:15:23.570
to be continuous so this is the approach why
do you go for designing the analysis everything
00:15:23.570 --> 00:15:28.040
take the y axis to be continuous x axis to
be discrete it becomes a discrete time only
00:15:28.040 --> 00:15:34.080
signal discrete time signal then after doing
your after getting your design and all that
00:15:34.080 --> 00:15:39.760
then carry out some further error analysis
where you try to find out the effect of this
00:15:39.760 --> 00:15:44.250
approximation on the y axis that y axis actually
is discrete but you assume to be continuous
00:15:44.250 --> 00:15:50.390
so what kind of error how much error does
it give raise to ok how to keep that error
00:15:50.390 --> 00:15:52.750
within limit within bound making small and
all that that is a separate chapter that is
00:15:52.750 --> 00:15:55.360
called finite for length analysis
00:15:55.360 --> 00:16:00.320
so this is the difference between discrete
time signal and digital signal now this one
00:16:00.320 --> 00:16:05.510
more region why this digital signal processing
or discrete type signal procedure here became
00:16:05.510 --> 00:16:11.770
very popular one is of course the digital
domain you can do any processing another is
00:16:11.770 --> 00:16:17.760
last thirty years or so we have seen tremendous
growth in digital v l s i not so much in analog
00:16:17.760 --> 00:16:26.210
v l s i but digital v l s i ok and that is
why is shifted from processing in analog domain
00:16:26.210 --> 00:16:29.480
to digital domain because in digital domain
you can do millions of operations you know
00:16:29.480 --> 00:16:36.790
so many gates and others you know can be and
a chip can be made so much of operation can
00:16:36.790 --> 00:16:44.950
be put in one eyes ok so that is why is shifted
to this ok these are the two main motivations
00:16:44.950 --> 00:16:49.340
i suggest some books also though i suggest
i tell you that be careful always follow my
00:16:49.340 --> 00:16:54.610
lectures because i normally don't follow any
book i teach from my understanding and here
00:16:54.610 --> 00:17:00.160
and there you will always find some new concepts
which i am not giving any book ok so that
00:17:00.160 --> 00:17:04.159
should be of value addition you should be
careful you should take note of this so the
00:17:04.159 --> 00:17:44.880
books are this by alan oppenheim and ron schafer
i don't remember the publisher these all but
00:17:44.880 --> 00:17:49.820
this two authors you know way back in meet
seventys when the d s p topic just first just
00:17:49.820 --> 00:17:56.420
came up they wrote a fantastic book that is
just digital signal processing i believe oppenheim
00:17:56.420 --> 00:18:02.180
schafer and there is another per third author
that i don't remember but this is a famous
00:18:02.180 --> 00:18:12.570
book but i personally like this book by the
same authors this is a fantastic book much
00:18:12.570 --> 00:18:17.850
shorter must thinner than that but this is
a fantastic book this this is more elaborate
00:18:17.850 --> 00:18:28.970
and sometimes unnecessarily elaborate unnecessarily
in the in buying opinion of course then there
00:18:28.970 --> 00:18:51.770
is a book d s p i don't remember there is
a title by proakins john proakins and mano
00:18:51.770 --> 00:19:01.150
full name i don't remember its title is manolakin
its a book a very good book then there is
00:19:01.150 --> 00:19:21.800
a d s p book by s k mitra this is also very
good this book they are pretty good they are
00:19:21.800 --> 00:19:26.980
very good at having very good examples and
everything but please follow my lectures because
00:19:26.980 --> 00:19:33.170
my lectures or for my lectures i don't follow
any of them i teach from my own understanding
00:19:33.170 --> 00:19:37.960
and here and there i give inputs which are
not easily found in book and you have to be
00:19:37.960 --> 00:19:39.540
careful for that
00:19:39.540 --> 00:19:46.460
so with this background i start this course
now so a starting point as i told you is a
00:19:46.460 --> 00:19:54.710
sequence sequence of numbers though this axis
is n there is no concept of time here thus
00:19:54.710 --> 00:20:00.740
compute i mean for processing we take only
the numbers even if the analog signal is quantized
00:20:00.740 --> 00:20:05.670
even if we started by [quantizing/sampling]
sampling an analog signal we take every sample
00:20:05.670 --> 00:20:11.010
by analog to digital converter then digitize
the sample see the sample height that the
00:20:11.010 --> 00:20:17.370
number we process the numbers ok this sample
so much value next sample so much value next
00:20:17.370 --> 00:20:22.880
sample those values those numbers numerical
values one upon another which is sequence
00:20:22.880 --> 00:20:28.540
of numerical values them this values only
i process that is one sample value so much
00:20:28.540 --> 00:20:33.830
another sample value so much another so much
another so much another so much another only
00:20:33.830 --> 00:20:40.420
the values i process sequence not so this
time of separation between this two sample
00:20:40.420 --> 00:20:46.190
is not my concern here ok the another digit
is that whatever algorithm i want to propose
00:20:46.190 --> 00:20:52.520
out of them that should be independent of
this sampling period that that is why we only
00:20:52.520 --> 00:20:57.210
take the numbers the sample heights and they
one after another they come and they form
00:20:57.210 --> 00:21:08.860
a sequence or denoted by say x n h n y n v
n so this is my x zero at zeroth point this
00:21:08.860 --> 00:21:17.120
is my x one this is my x two this is my x
three and dot dot dot i can have on this side
00:21:17.120 --> 00:21:26.430
also this is my x minus one this is my x minus
two because this is zero this is one this
00:21:26.430 --> 00:21:30.750
is two this is minus one minus two dot dot
dot these are typical sequence sequence of
00:21:30.750 --> 00:21:37.310
numbers it could have come by sampling an
analog signal so and then i just take the
00:21:37.310 --> 00:21:42.351
sample values the train of samples one after
another i don't care for the time of separation
00:21:42.351 --> 00:21:45.700
there is a sequence who is after whom who
is before whom who is zeroth who is first
00:21:45.700 --> 00:21:50.150
who is second like that or this could be generated
just as a sequence of numbers manually by
00:21:50.150 --> 00:21:56.690
myself or by a computer which is generating
data i think like that ok
00:21:56.690 --> 00:22:07.690
this is a sequence sequence is there are some
properties first if i give you one sequence
00:22:07.690 --> 00:22:49.220
x one n something like this something like
this dot dot dot and x two n
00:22:49.220 --> 00:23:05.770
x two n dot dot dot dot dot when i add x one
n plus x two n by this i mean another sequence
00:23:05.770 --> 00:23:18.690
y n where
y n could be like this zeroth sample so this
00:23:18.690 --> 00:23:25.030
is a zeroth sample and zeroth sample this
two will be added so new zeroth sample sorry
00:23:25.030 --> 00:23:30.560
this is zeroth sample and this is zeroth sample
this two will be added new zeroth sample so
00:23:30.560 --> 00:23:42.610
if it is a if it is b if it is b mu value
will be a plus b ok so sample wise addition
00:23:42.610 --> 00:23:54.540
let me draw a better figure this is what clumsy
so if it is a sequence like this say a b c
00:23:54.540 --> 00:24:11.179
dot dot dot dot and i call it x one n and
another sequence say a prime b prime c prime
00:24:11.179 --> 00:24:24.260
dot dot dot dot and i call it x two n then
y n equal to x one n plus x two n this will
00:24:24.260 --> 00:24:30.610
be thus we have to add sample wise zeroth
sample of this fellow zeroth sample of this
00:24:30.610 --> 00:24:35.960
fellow they have to be added that will be
new zeroth sample so this will be a plus a
00:24:35.960 --> 00:24:44.090
prime a plus a prime together is that you
get the new zeroth sample at zeroth point
00:24:44.090 --> 00:24:51.160
similarly point number one at index one b
and b prime so we add at index one there will
00:24:51.160 --> 00:24:59.010
be new value b plus b prime at index two again
you have got c c prime so you add the two
00:24:59.010 --> 00:25:05.360
values you will get c plus c prime at index
two like that so at every index take the sample
00:25:05.360 --> 00:25:08.990
here take the sample here at the two that
will be the new sample at the same index this
00:25:08.990 --> 00:25:13.040
is very simple this is called addition
00:25:13.040 --> 00:25:21.170
then multiplying a sequence say x one n you
have to multiply by say um say beta beta into
00:25:21.170 --> 00:25:28.960
x one is nothing every sample will be multiplied
by beta so zeroth sample earlier was a now
00:25:28.960 --> 00:25:37.960
it will be beta a next sample was b so it
will be beta b next sample was c now it will
00:25:37.960 --> 00:25:46.920
be beta c ok dot dot dot dot so scalar multiplication
of every sample means multiplying by every
00:25:46.920 --> 00:25:52.030
sample that is beta times a sequence means
in a scalar times a sequence means take every
00:25:52.030 --> 00:25:59.720
sample multiplied by the scalar each of them
multiplied by the scalar ok then comes a very
00:25:59.720 --> 00:26:17.170
important thing shifting delaying first what
is shifting suppose you have been given x
00:26:17.170 --> 00:26:36.540
of n it has got a value at zeroth point may
be a then b then c dot dot dot dot ok at minus
00:26:36.540 --> 00:26:46.360
one it has got say may be a prime at minus
two it has got say b prime dot dot dot dot
00:26:46.360 --> 00:27:01.300
suppose this is a sequence if i have y n is
equal to x n minus one its n minus one if
00:27:01.300 --> 00:27:08.179
n is an integer n minus x is a function of
some integer right x zero has a value x one
00:27:08.179 --> 00:27:12.870
has a value x two has a value everywhere given
so if n is an integer n minus one also is
00:27:12.870 --> 00:27:18.210
an integer so x of that also has a value from
here only i can pick up
00:27:18.210 --> 00:27:28.200
so but how will this look like then so for
that what is y one suppose i said example
00:27:28.200 --> 00:27:35.420
you take y one y one will be x of one minus
one zero so at index one there is why this
00:27:35.420 --> 00:27:46.179
i am plotting y n so at y one value will be
x zero so this fellow will move here so it
00:27:46.179 --> 00:27:54.970
is moving to the right what is y two it will
be x two minus one x one so this was x one
00:27:54.970 --> 00:28:05.040
this will now move here y two so this is moving
to the right this is moving to the right what
00:28:05.040 --> 00:28:13.960
will be y zero y zero is x minus one that
is this guy a prime so a prime is also moving
00:28:13.960 --> 00:28:21.950
from left so it is shifting so this sequence
is just shifted to the right by one ok if
00:28:21.950 --> 00:28:31.050
it is n minus one if it is y n is equal to
x n minus say k where k is an integer how
00:28:31.050 --> 00:28:40.890
will it look like
start with y n equal to k n equal to k means
00:28:40.890 --> 00:28:51.350
k minus k zero so at n equal to k n equal
to k y n that is y k will be x zero so x zero
00:28:51.350 --> 00:28:58.240
from here into a this sample value will move
here so it is making a jump from zeroth index
00:28:58.240 --> 00:29:07.000
to k th index it is ok then y k plus one if
i put k plus one it will be k plus one minus
00:29:07.000 --> 00:29:20.900
k so on x one from location one will move
to location k plus one ok y k plus two y k
00:29:20.900 --> 00:29:27.220
plus two will be x two so this this guy you
see will now move here so you see they are
00:29:27.220 --> 00:29:36.350
jumping from zeroth to k th from one th to
k plus one th from two th to k plus two th
00:29:36.350 --> 00:29:43.730
so this sequence is shifted to the right by
k if i write x n minus k ok
00:29:43.730 --> 00:29:56.830
on the other hand
on the other hand if i have y n is equal to
00:29:56.830 --> 00:30:09.390
x n plus k then what will happen we will start
at n equal to minus k y minus k will be x
00:30:09.390 --> 00:30:15.860
minus k plus k that is zero x of zero what
was x of zero a so this a from here will move
00:30:15.860 --> 00:30:26.679
here to the left side so it is moving to the
left at minus k plus one minus k plus one
00:30:26.679 --> 00:30:36.710
it is further minus y of minus of k plus one
if you put here you get x minus one sorry
00:30:36.710 --> 00:30:46.390
same from come here first i will go to there
later minus of k one less k minus one say
00:30:46.390 --> 00:30:53.240
it was minus k to the right will be one less
so minus of k minus one y of minus of k minus
00:30:53.240 --> 00:30:59.100
one if you put that here at this point you
will see it will become x one so this guy
00:30:59.100 --> 00:31:11.780
he will come here so a is moving to the left
by k b is moving to the left by k so even
00:31:11.780 --> 00:31:16.450
that way you can verify every circle is moving
to the left by k that is sequence is shifted
00:31:16.450 --> 00:31:22.080
to the left we call it advanced by k so if
k is negative sequence will be shifted to
00:31:22.080 --> 00:31:27.050
the right by k if k is positive it will be
shifted to the left by k the shift has something
00:31:27.050 --> 00:31:33.140
to do with delaying if it is a real time signal
that is if this sample points corresponds
00:31:33.140 --> 00:31:39.799
to time axis points then it amounts to delaying
or advancing that i am coming to in the next
00:31:39.799 --> 00:31:40.799
module
00:31:40.799 --> 00:31:40.929
thank you