Sampling Theorem

Summary

This lecture delves into the concept of sampling and the Sampling Theorem, crucial for understanding the transition from continuous-time to discrete-time signals in digital systems. It explains that real-life signals are analog, but digital systems require their conversion to discrete signals. The process involves multiplying a band-limited continuous-time signal by a periodic impulse train, creating a sampled signal. The lecture emphasizes the importance of the sampling frequency being at least twice the maximum frequency component of the original signal to avoid overlapping in the frequency domain, which is essential for accurate signal recovery. The Sampling Theorem is encapsulated as the condition for signal recovery from its samples, underlining the significance of using band-limited signals to prevent overlapping and ensure accurate signal reconstruction.

Takeaways

• 📚 The lecture introduces the concept of sampling and the Sampling Theorem, focusing on the transition from continuous-time to discrete-time signals.
• 🌐 The necessity to study discrete-time signals arises from the extensive use of digital technologies, which require conversion of analog, continuous-time signals to a format they can process.
• 🔍 Sampling is defined as the process of converting a continuous-time signal into a discrete-time signal, which is essential for digital systems to handle analog information.
• 📶 The script assumes the use of band-limited signals for sampling, which have a finite range of frequencies in their Fourier transform, simplifying the reconstruction process.
• 🔢 The maximum frequency component of the message signal is denoted as Omega M and plays a crucial role in determining the conditions for signal reconstruction.
• 🔄 The sampling process involves multiplying the continuous-time signal by a periodic impulse train, represented as CT, which has a fundamental period equal to the sampling interval T_s.
• 📈 The Fourier transform of the sampled signal, S(Ω), can be found by convolving the Fourier transforms of the message signal M(Ω) and the impulse train C(Ω).
• 🔁 The spectrum of the sampled signal consists of repeated images of the message signal's spectrum, shifted by integer multiples of the sampling frequency Omega S.
• 🚫 The condition for accurate signal recovery from its samples is that the sampling frequency (Omega S) must be greater than twice the maximum frequency component of the message signal (Omega M), preventing spectral overlap.
• 📉 If the sampling frequency is less than twice the maximum frequency, overlapping occurs in the frequency domain, making it impossible to accurately recover the original continuous-time signal.
• 📚 The Sampling Theorem concludes that a band-limited signal can be fully represented by its samples and reconstructed when the sampling frequency meets the specified condition.

Q & A

What is the primary reason we study discrete-time signals despite having continuous-time signals?

The primary reason is the extensive use of digital technologies in today's world. Since all real-life signals are analog or continuous in nature, and digital systems cannot process them directly, it is necessary to convert continuous-time signals to discrete-time signals for processing.

Define sampling in the context of signal processing.

Sampling is the process of reducing a continuous-time signal to a discrete-time signal, which can be processed by digital systems.

Why is it important to consider band-limited signals when discussing sampling?

Band-limited signals are important because their Fourier transform or spectrum is nonzero only within a finite range of frequencies. This property is crucial for accurately extracting the continuous-time signal from the sampled discrete-time signal.

What is the significance of the maximum frequency component (ΩM) of a message signal?

ΩM is significant because it represents the highest frequency component of the message signal. It is used to determine the relationship with the sampling frequency (ΩS) and to ensure that the sampled signal can be accurately recovered from its samples.

What is a sampler in the context of signal processing?

A sampler is a device that acts as a multiplier, taking a continuous-time signal and a periodic impulse train as inputs, and producing a sampled signal as output.

What is the relationship between the sampling period (Ts) and the sampling frequency (ΩS)?

The sampling frequency (ΩS) is the reciprocal of the sampling period (Ts). It is calculated as ΩS = 2π/Ts.

How does the spectrum of the sampled signal (ST) relate to the spectrum of the message signal (MT) and the impulse train (CT)?

The spectrum of the sampled signal (ST) is obtained by convoluting the spectrum of the message signal (MT) with the spectrum of the impulse train (CT) and then dividing by 2π.

What is the condition for accurately recovering the continuous-time signal from its sampled signal?

The condition for accurate recovery is that the sampling frequency (ΩS) must be greater than or equal to twice the maximum frequency component (ΩM) of the message signal.

What is the sampling theorem?

The sampling theorem states that a signal can be represented by its samples and accurately recovered if the sampling frequency is greater than or equal to twice the maximum frequency component of the signal.

What happens when the sampling frequency is less than twice the maximum frequency component of the signal?

When the sampling frequency is less than twice the maximum frequency component, overlapping occurs in the frequency spectrum, which prevents accurate recovery of the continuous-time signal from its samples.

Why is it necessary to avoid overlapping in the frequency spectrum when sampling a signal?

Avoiding overlapping is necessary to ensure that each frequency component of the original signal can be uniquely identified in the sampled signal, allowing for accurate recovery of the original signal from its samples.

What is the role of the guard band in the frequency spectrum of a sampled signal?

The guard band is the gap between the shifted versions of the message signal's spectrum in the sampled signal. It ensures that there is no overlapping, which is crucial for the accurate recovery of the original signal from its samples.

Outlines

00:00

📚 Introduction to Sampling and Sampling Theorem

This paragraph introduces the concept of sampling and the sampling theorem. It explains the necessity of studying discrete-time signals in the digital era, despite the prevalence of continuous-time signals in real life. The process of converting a continuous-time signal to a discrete-time signal is defined as sampling. The importance of band-limited signals is highlighted, as their Fourier transform is nonzero only within a finite range of frequencies. This property is crucial for the reconstruction of the original signal from its samples. The concept of the maximum frequency component (Ω_M) is introduced, which will be frequently used throughout the course.

05:03

🔢 Sampling Process and the Resultant Signal

The second paragraph delves into the process of sampling, where a continuous-time signal is multiplied by a periodic impulse train to create a sampled signal. The properties of the impulse train are described, including its fundamental time period (T_s) and the associated sampling frequency (Ω_s). The multiplication of the message signal (M_T) with the impulse train results in a new signal, where each impulse carries the weight of the instantaneous value of M_T at that specific time. The Fourier transform of the sampled signal is then discussed, emphasizing the convolution of the Fourier transforms of the message signal and the impulse train.

10:04

📈 Fourier Transform of the Sampled Signal

This paragraph focuses on obtaining the Fourier transform of the sampled signal. It outlines the convolution properties used to simplify the expression for the Fourier transform of the sampled signal. The result is an expansion that includes the message signal's spectrum shifted by integer multiples of the sampling frequency (Ω_s). The discussion leads to a visual representation of the spectrum, emphasizing how the spectrum appears when there is no overlapping of the shifted signal components.

15:06

🔑 The Sampling Theorem and Its Conditions

The fourth paragraph explains the conditions under which the original continuous-time signal can be accurately recovered from its samples. It describes the relationship between the sampling frequency (Ω_s) and the maximum frequency component of the message signal (Ω_M). The sampling theorem states that accurate recovery is possible when the sampling frequency is greater than or equal to twice the maximum frequency of the signal. Two scenarios are discussed: one where there is a gap between the shifted signal components, and another where the components touch without overlapping. The possibility of overlapping, which prevents accurate recovery, is also explained.

20:08

🚫 Limitations of Non-Band-Limited Signals

The final paragraph addresses the limitations when dealing with non-band-limited signals. It emphasizes that such signals cannot be accurately recovered from their samples due to overlapping in the frequency domain. The importance of considering band-limited signals to avoid overlapping and ensure signal recovery is reiterated. The lecture concludes with a summary of the key points and a look forward to the next lecture.

Keywords

💡Sampling

Sampling is the process of converting a continuous-time signal into a discrete-time signal, which is a fundamental concept in the video. It is essential for understanding how analog signals are transformed into a format that can be processed by digital systems. The script mentions that 'sampling is the process of reduction of a continuous time signal to a discrete time signal', illustrating its importance in the context of digital signal processing.

💡Sampling Theorem

The Sampling Theorem is a key principle discussed in the video that dictates the conditions under which a continuous-time signal can be accurately recovered from its samples. It states that the sampling frequency must be at least twice the maximum frequency component of the signal. The script refers to this theorem when explaining that 'a signal can be represented in its samples... and can be recovered back when the sampling frequency is greater than or equal to twice of maximum frequency component present in the signal'.

💡Continuous-Time Signal

A continuous-time signal is a type of signal that can take on any value at any point in time and is naturally occurring in real life. In the video, it is mentioned that 'all real life signals are analog in nature or you can say all real life signals are continuous in nature', emphasizing the prevalence of continuous-time signals in everyday analog systems.

💡Discrete-Time Signal

A discrete-time signal is a signal that is sampled at discrete time intervals and is used in digital systems. The script explains the transition from continuous-time to discrete-time signals as 'the process of conversion or you can say, reduction of continuous time signal to discrete time signal is known as sampling', highlighting the transformation necessary for digital processing.

💡Digital Technologies

Digital technologies are electronic systems that process information in discrete form rather than as a continuous signal. The video script discusses the 'extensive use of digital technologies nowadays', indicating why the study of discrete-time signals is relevant, as these technologies are prevalent in modern society.

💡Band-Limited Signal

A band-limited signal is one whose frequency spectrum is confined within a finite range of frequencies. The script defines it by stating that 'band limited signal means its Fourier transform... will be nonzero only within a small range of frequencies'. This property is crucial for the sampling process because it prevents overlapping in the frequency domain, which is necessary for signal recovery.

💡Fourier Transform

The Fourier Transform is a mathematical technique used to convert a time-domain signal into its frequency-domain representation. In the context of the video, it is used to analyze the properties of the continuous-time signal, as mentioned when the script refers to 'the Fourier transform or you can say the spectrum of the signal'.

💡Maximum Frequency Component (Omega M)

Omega M represents the highest frequency component of a message signal in the video. It is a critical parameter in the Sampling Theorem, as the script explains that 'Omega M will be used a lot in this course so remember what it is, it is the maximum frequency component of the message signal'. It helps determine the sampling frequency required for signal recovery.

💡Sampling Frequency (Omega S)

Omega S, or the sampling frequency, is the frequency at which a continuous-time signal is sampled in a digital system. The script describes it as 'the angular frequency of this periodic impulse train' and emphasizes its relationship with the maximum frequency component, stating that 'we are going to get relation between these two frequencies only'.

💡Periodic Impulse Train

A periodic impulse train is a series of equally spaced delta functions that serve as the sampling function in signal processing. The script describes it as 'this periodic impulse train is having the fundamental time period equal to Ds', which is used to sample the continuous-time signal and create the discrete-time signal.

Highlights

Introduction to sampling and the sampling theorem in the context of transitioning from continuous-time to discrete-time signals.

The necessity of studying discrete-time signals due to the prevalence of digital technology in processing analog signals.

Definition of sampling as the process of converting a continuous-time signal to a discrete-time signal.

Condition for recovering the continuous-time signal from the discrete-time signal.

Assumption of the message signal being band-limited for the sampling process.

Explanation of a band-limited signal and its importance in avoiding overlapping in the frequency domain.

Importance of Omega M, the maximum frequency component of the message signal, in the sampling process.

Introduction of the sampler device and its function as a multiplier in the sampling process.

Description of the periodic impulse train and its relation to the sampling frequency.

The process of obtaining the spectrum of the sampled signal through Fourier transform.

Convoluting the Fourier transform of the message signal with that of the impulse train.

The relationship between the sampling frequency (Omega s) and the maximum frequency component (Omega M).

Conditions for avoiding overlapping in the frequency spectrum and the concept of the guard band.

The critical condition for signal recovery: the sampling frequency must be greater than or equal to twice the maximum frequency component.

Illustration of the consequences of not meeting the sampling theorem condition, leading to overlapping and signal recovery issues.

The final statement of the sampling theorem and its significance in digital signal processing.

Example to demonstrate the importance of considering band-limited signals to prevent overlapping in the frequency domain.

Conclusion of the lecture with a summary of the key points and an invitation to the next lecture.

Transcripts

in this lecture we will understand what

do we mean by sampling and what is

sampling theorem till now we have done a

lot of discussions on continuous-time

signals and now it's time to move on to

discrete-time signals but wait why we

are required to study the concepts

related to discrete-time signals when we

already have continuous-time signals the

answer is extensive use of digital

technologies nowadays so if you look

around yourself you will find there are

so many digital technologies in

implementation but we know all real life

signals are analog in nature or you can

say all real life signals are continuous

in nature so they are continuous time

signals so the information is carried by

continuous time signal but the systems

we are using our digital in nature

therefore they will not process

continuous time signal and it becomes

important to convert the continuous time

signal to discrete time signal and this

process of conversion or you can say

reduction of continuous time signal to

discrete time signal is known as

sampling now let us define sampling

properly sampling is the process of

reduction of a continuous time signal to

a discrete time signal and if you want

to get the continuous time signal back

from the discrete time signal then you

can do this but there is some condition

and now we will try to obtain that

particular condition so let's move on to

the next part of this lecture let us

take one continuous time signal and

let's say we want to convert this

continuous time signal to a discrete

time signal and we are representing this

continuous time signal by M T and

representing it by empty because this

signal is known as message signal this

is known as message signal because this

signal is carrying the information we

want to process and you must note one

point here this signal we have taken is

a band limited signal we are assuming

that we have taken a continuous-time

signal which is a band limited signal

now why we are taking band limited

signal and what is band limited signal I

will explain now band limited signal

means its Fourier transform or you can

say the spectrum of the signal will be

nonzero only within a small range of

frequencies for example let's say this

signal is having the Fourier transform M

Omega like this now you can see that the

Fourier transform or the spectrum is non

zero from minus Omega M 2 plus Omega M

so here in this case we are having the

spectrum which is nonzero only for a

finite range of frequencies and it will

be very helpful when we will try to

extract the continuous-time signal from

the converted discrete-time signal and

this point will be more clear after some

time so for now just understand that we

are having a continuous-time signal

carrying the message and his Fourier

transform is like this and the Fourier

transform is nonzero between minus Omega

M and Omega M and also note down what is

Omega M Omega M is the maximum maximum

frequency component of the message

signal MT Omega M will be used a lot in

this course so remember what it is it is

the maximum frequency component of

empty if empty is having multiple

frequency components then the maximum

frequency out of those frequency

components is Omega M in the next

lecture we will solve numerical problem

and there this point will be more clear

now we will move on to the next process

and the next process is feeding this

continuous-time signal to a device known

as sampler now this device sampler is

simply acting as multiplier it is acting

as multiplier and multiplier means it is

going to multiply to continuous-time

signals first signal is empty and the

second signal is CT and you can clearly

see that Ct is a periodic impulse train

and this periodic impulse train is

having the fundamental time period equal

to D s so T s is the fundamental time

period or it is the sampling period it

is also known as sampling period it is

also known as sampling interval it is

also known as sampling interval and we

know Omega s which is the angular

frequency is equal to 2 pi divided by T

s and Omega s is known as the sampling

frequency it is known as the sampling

frequency so now we have two frequencies

the first frequency is Omega M known as

the maximum frequency component of MT

and the another frequency we have is

Omega s which is the sampling frequency

or the angular frequency of this

periodic impulse train and remember both

the frequencies because we are going to

get relation between these two

frequencies only and as we are having

the periodic impulse train like this we

can write it as summation and equal to

minus infinity to infinity Delta t minus

n TS now when n is equal to 0 we will

have Delta T this means impulse present

at the origin when n is equal to 1 we

will have Delta t minus TS this means

the unit impulse which is present at TS

simply shift the unit impulse present at

the origin towards the right by TS and

you will have delta t minus t s so this

impulse you are looking here is delta t

minus TS this impulse is delta T this

impulse is delta t plus TS and we will

have it when n is equal to minus 1

similarly we will have all the other

impulses and we will have the impulse

train like this now after multiplying

empty and CT we will have the resultant

waveform like this in this waveform we

are having different impulses and the

area or weight of the impulse is equal

to the instantaneous value of signal MT

for example if you talk about the

impulse present at T s then this impulse

will have the weight or strength equal

to the instantaneous value of M T this

means when T is equal to T s the value

of M T will be the strength or weight of

this particular impulse and in this way

we have a new signal which is known as

sampled signal and represented by as T s

T is equal to empty message signal

multiplied to the periodic impulse train

C T and now we are interested in

obtaining the spectrum of the sampled

signal this means we are having s T and

we want to have the Fourier transform as

Omega of

steal and we also know that s T is equal

to message signal multiplied to the

periodic impulse train CT and we know

the property of Fourier transform when

two signals are multiplied if you try to

take the Fourier transform then on the

left hand side you will have the Fourier

transform of this signal which is s

Omega and on the right hand side you

will have 1 divided by 2 pi multiplied

to the Fourier transform of Mt which is

M Omega convoluted with the Fourier

transform of CT and let's say Fourier

transform of CD

is C Omega so we can easily obtain the

Fourier transform as Omega after

convoluting M Omega and C Omega and then

dividing it by 2 pi M Omega we have

assumed here but what about C Omega if

you remember in the Fourier transform

solved problem number 11 we solved one

problem in which the time domain signal

was exactly like this and we obtained

the Fourier transform of this signal and

the result we got was equal to Omega s

summation n equal to minus infinity to

infinity Delta Omega minus n Omega s so

we have C Omega and we have M Omega now

let's move on to the simplification of

this and in order to simplify this we

need to use the properties of

convolution and the properties of

impulse signal we are having as Omega as

Omega equal to 1 over 2 pi 1 over 2 pi

multiplied to M Omega M Omega

convolution with C Omega and Co

is equal to this so we will write Omega

as summation and equal to minus infinity

to infinity Delta Omega minus and Omega

s and we can take this Omega s outside

then we will have s Omega equal to Omega

s divided by 2 pi inside the bracket M

Omega convolution with summation and

equal to minus infinity to infinity

Delta Omega minus and Omega s and we

know Omega s divided by 2 pi will be

equal to 1 over TS from here you can see

that Omega s divided by 2 pi will be

equal to 1 over TS

so in the next step we will write Omega

s divided by 2 pi equal to 1 over TS and

inside the bracket we can rearrange this

and we can write summation n equal to

minus infinity to infinity M Omega M

Omega convolution who with Delta Omega

minus n Omega s and we know the property

by which we can simplify this further

and according to the property if we have

a signal XT and the signal is convoluted

with Delta t minus T 1 then this is

equal to X t minus T 1 we simply need to

put t minus t 1 in place of T and if we

follow this we will put Omega minus n

Omega s in place of this Omega so in the

next step we will have s Omega equal to

1 divided by TS inside the bracket

summation and equal to minus infinity to

infinity M Omega minus n Omega s M Omega

minus n Omega s this is what we need and

now let X

and this by putting the different values

of n so we will have as Omega equal to 1

divided by TS inside the bracket when n

is equal to 0 we will have M Omega we

will have M Omega and we have the wave

form of M Omega now when n is equal to 1

we will have M Omega minus Omega s so we

will have M Omega minus Omega s

similarly we will have different terms

when n is equal to minus 1 we will have

ham Omega plus Omega s we will have M

Omega plus Omega s similarly for

negative values of n we will have

different terms so in this way we are

getting the expansion of has Omega and

using this expansion we can plot the

wave form of s Omega and the wave form

will look like this if you refer the

expansion you will find when n is equal

to 0 we are having M Omega this means we

should have this wave form and therefore

this wave form here is the wave form of

M Omega this makes this frequency equal

to minus Omega m and this frequency

equal to Omega M and when n is equal to

1 we are having M Omega minus Omega s

this means M Omega has shifted towards

the right by Omega s this makes this

frequency here this frequency equal to

Omega s minus Omega M this frequency

here equal to Omega M plus Omega s or

you can write Omega S Plus Omega M and

this particular frequency here equal to

Omega s now you can notice one thing

that Omega s minus Omega M is greater

than Omega M

Omega s minus Omega M is greater than

Omega M this implies we are getting

omega s greater than twice of Omega M so

when Omega S which is the sampling

frequency greater than twice of maximum

frequency component of the message

signal then there is no overlapping in

this spectrum by no overlapping I mean

this triangle and this triangle are not

overlapping there is sufficient gap

between the two triangles and this gap

here this gap here is known as guard

band now what will happen when Omega s

minus Omega m is equal to Omega M this

means this frequency here which is Omega

M is equal to this frequency here which

is omega s minus Omega M it is very

obvious that the triangles will be

touching so you will have the waveform

like this when Omega s minus Omega M is

equal to Omega M or you can say Omega s

is equal to twice of Omega M now in the

two scenarios we have discussed till now

we can recover the continuous-time

signal from the sampled signal because

there is no overlapping in the first

case there is a gap and hence there is

no overlapping in the second case they

are touching and again there is no

overlapping but if there is overlapping

then we cannot recover the

continuous-time signal accurately now

let's try to understand in what

condition the overlapping will occur we

have discussed the first case in which

we found when Omega s is greater than

twice of Omega

there will be gap and when Omega s is

equal to twice of Omega M there will be

touching case and when Omega s is less

than twice of Omega M

when Omega s is less than twice of Omega

M then there will be overlapping as you

can see on your screen and you can

understand the point that Omega s minus

Omega is less than Omega M this makes

the overlapping possible and in this

scenario you cannot recover the

continuous-time signal accurately and

this is known as sampling theorem so

let's move on to the final statement of

sampling theorem

according to sampling theorem a signal

can be represented in its samples like

we have done we have represented signal

MT in its samples which is sampled

signal st and can be recovered back when

the sampling frequency this means omegas

is greater than or equal to twice of

maximum frequency component present in

the signal this means twice of Omega M

so when omegas is greater or equal to

twice of Omega M we can recover back the

signal from its samples so remember this

condition omegas should be greater or

equal to twice of Omega M this is a very

important condition so this is all for

the sampling theorem and initially I

told you that we are considering the

band limited signal we considered our

message signal to be band limited signal

to get before you transform like this

and we are getting the Fourier transform

like this because we don't want the

overlapping now we will try to

understand this point by the help of one

example and in this example we are

considering the time domain signal has

not band-limited and as the time domain

signal is not band limited the Fourier

transform will not be bounded like this

it will be something like this now

repeat this structure like this and you

will find there is over lapping and we

are trying to avoid the overlapping so

that we can recover the time domain

signal from its samples but if you take

the not band-limited signal you will

have the overlapping and therefore

recovery will not be possible and this

is the reason we are taking the

band-limited

signal and this is all for this lecture

see you in the next one

[Applause]

[Music]