Reconstruction of a Signal Using Practical LPF
Summary
TLDRIn this lecture, the concept of signal recovery using practical and ideal low-pass filters is discussed, with a focus on critical sampling. The lecturer explains that while an ideal low-pass filter can recover a signal when the sampling frequency is twice the maximum message frequency, a practical low-pass filter introduces limitations due to the transition band. This results in incomplete recovery of the message signal. The lecture concludes that signal recovery under critical sampling conditions with a practical low-pass filter is not possible, and includes a homework question on the optimal sampling mode.
Takeaways
- đ§ The lecture focuses on using a practical low-pass filter to recover a message signal under critical sampling conditions.
- đ Critical sampling occurs when the sampling frequency (Omega s) equals twice the maximum frequency component of the message signal (Omega M).
- đŻ Ideal low-pass filters can successfully recover a message signal if the critical frequency equals the maximum frequency component of the message signal.
- đïž Practical low-pass filters, unlike ideal filters, have a gradual transition from pass band to stop band, which introduces a transition band.
- đ In the case of practical low-pass filters, some unwanted portions of the sampled signal are passed along with the desired signal, leading to signal distortion.
- đ« With practical low-pass filters, it is not possible to perfectly recover the message signal in critical sampling conditions.
- â The lecture concludes that option B (false) is correct: recovery is not possible using a practical low-pass filter when Omega s is twice Omega M.
- đ The Fourier transform is essential in analyzing the input signal and determining whether the recovered signal matches the original message signal.
- đ§ For homework, students are asked to determine which sampling mode should be used with a practical low-pass filter: critical sampling, undersampling, or oversampling.
- đ The importance of understanding the frequency response of practical filters, as it impacts the ability to recover the original signal, is emphasized.
Q & A
What is the primary focus of the current lecture?
-The primary focus is on determining whether the message signal can be recovered using a practical low-pass filter when the sampling frequency (Omega s) is equal to twice the maximum frequency component of the message signal (Omega M).
What is critical sampling, and how is it defined?
-Critical sampling occurs when the sampling frequency (Omega s) is exactly twice the maximum frequency component (Omega M) of the message signal. Under this condition, the spectrums of the signal touch each other.
What are the key differences between an ideal and a practical low-pass filter?
-An ideal low-pass filter has a sharp cutoff between pass band and stop band, while a practical low-pass filter has a transition band where the transition from pass to stop band is gradual rather than instantaneous.
What is the role of the pass band in a low-pass filter?
-The pass band is the frequency range between -Omega C and Omega C, where the low-pass filter allows the signal to pass through. Only the signal components within this frequency range appear at the output.
What happens in the stop band of a low-pass filter?
-In the stop band, the low-pass filter blocks the input signal, meaning any signal components with frequencies greater than Omega C are not passed to the output.
Why is it important to match the cutoff frequency (Omega C) to the maximum frequency component (Omega M) in the ideal filter case?
-When the cutoff frequency (Omega C) is equal to the maximum frequency component of the message signal (Omega M), the signal can be perfectly recovered, as the entire message signal spectrum falls within the pass band.
Can the message signal be perfectly recovered using a practical low-pass filter under critical sampling?
-No, the message signal cannot be perfectly recovered using a practical low-pass filter under critical sampling because the transition band of the filter causes additional unwanted components to pass through, leading to distortion.
What is the effect of the transition band in a practical low-pass filter?
-The transition band causes some components outside the desired pass band to pass through, leading to incomplete signal recovery and distortion in the recovered signal.
Why is the answer to the question in the lecture 'false'?
-The answer is 'false' because with a practical low-pass filter, the message signal cannot be fully recovered under the condition Omega s = 2 * Omega M, due to the presence of the transition band in the filter's response.
What homework question is posed at the end of the lecture?
-The homework question asks which sampling mode should be used with a practical low-pass filter: (a) critical sampling, (b) under-sampling, or (c) over-sampling. Students are encouraged to post their answers in the comments, along with any relevant conditions.
Outlines
đ§ Introduction to Signal Reconstruction with Practical Low-Pass Filters
This paragraph introduces the transition from ideal to practical low-pass filters in signal reconstruction. It outlines the objective of the lecture, which is to determine whether a practical low-pass filter can recover the message signal under the condition that the sampling frequency (Omega s) is equal to twice the maximum frequency component (Omega M) of the message signal. The paragraph explains the concept of critical sampling and emphasizes the differences between ideal and practical low-pass filters.
đ Understanding the Pass Band and Stop Band of Low-Pass Filters
The second paragraph delves into the concepts of pass band and stop band in low-pass filters. It explains how a low-pass filter only passes signal components within a certain frequency range (pass band) and blocks components outside this range (stop band). This detailed description clarifies how the frequency response of a practical low-pass filter works and how these frequency bands impact the signal reconstruction process.
đ Fourier Transform and Signal Recovery
This paragraph focuses on how the input signalâs Fourier transform interacts with the frequency response of a low-pass filter. It explains the process of obtaining the recovered message signal by multiplying the Fourier transform of the input signal with the filterâs frequency response. The explanation emphasizes how this process leads to the recovered signal, which may or may not match the original message signal depending on certain conditions.
âïž Critical Sampling with an Ideal Low-Pass Filter
Here, the lecture explores the case of critical sampling with an ideal low-pass filter. By explaining the multiplication of the frequency response (H Omega) with the signalâs Fourier transform (S Omega), the paragraph demonstrates that when the cutoff frequency equals the maximum frequency of the message signal, the signal is perfectly recovered. It concludes that under these conditions, the recovered message signal is identical to the original.
â Practical Low-Pass Filter and Signal Recovery Challenges
This paragraph explains why signal recovery fails when using a practical low-pass filter with critical sampling. It highlights the existence of a transition band in practical filters, which causes additional unwanted signal components to pass through. This results in an imperfectly recovered signal that does not match the original message signal. As a result, the lecture concludes that signal recovery is not possible under these conditions.
đ Homework: Exploring Sampling Modes with Practical Filters
The final paragraph presents a homework assignment. It challenges the reader to determine the appropriate sampling mode when using a practical low-pass filter. The three options given are critical sampling, undersampling, and oversampling. Students are encouraged to post their answers in the comments, along with any conditions that apply to their choice.
Mindmap
Keywords
đĄLow-pass filter
đĄPractical low-pass filter
đĄIdeal low-pass filter
đĄOmega s
đĄOmega M
đĄCritical sampling
đĄPass band
đĄStop band
đĄTransition band
đĄFourier transform
Highlights
Reconstruction of signals using a low-pass filter, differentiating between ideal and practical filters.
Introduction to critical sampling: when sampling frequency is twice the maximum frequency component.
Explanation of pass band in a low-pass filter, which passes the portion of input signals between the cutoff frequencies.
Explanation of stop band, where the input signal frequencies are stopped by the filter when they exceed the cutoff frequency.
Process of recovering message signals using an ideal low-pass filter and how it relates to sampling.
The importance of matching the critical frequency with the maximum frequency component of the message signal for successful recovery.
Fourier transform of the input signal and how it interacts with the filter's frequency response to recover the signal.
Ideal low-pass filter successfully recovers the message signal when the critical frequency equals the maximum frequency.
Discussion of practical low-pass filters: differences in frequency response compared to ideal filters.
Introduction of the transition band in practical low-pass filters, which affects signal recovery.
In critical sampling with practical low-pass filters, parts of the spectrum that shouldn't be passed are passed due to the transition band.
Conclusion: Recovery of the original message signal is not possible with a practical low-pass filter under critical sampling conditions.
Answer to the question: the correct option is 'false' for recovering signals under critical sampling with a practical low-pass filter.
Homework problem posed: determining which sampling mode (critical, under, or over) should be used with a practical low-pass filter.
Instructions to submit the answer to the homework problem in the comments, with any conditions if applicable.
Transcripts
in the last lecture we saw how to
reconstruct his signal using a low-pass
filter and in this lecture we will try
to answer a question according to the
question who with practical low-pass
filter can we recover the message signal
when Omega s is equal to twice of Omega
M there are two options a is true and B
is false so from the question it is
clear that we are using a practical
low-pass filter and in the previous
lecture we used an ideal low-pass filter
so instead of ideal low-pass filter we
will use the practical low-pass filter
and therefore we must have the knowledge
of frequency response of the practical
low-pass filter and then we need to
focus on this condition Omega s is equal
to twice of Omega M this means sampling
frequency is equal to two times the
maximum frequency component of the
message signal and we know under this
condition the waveform of s Omega will
look like this the spectrums will be
touching each other and this particular
condition is known as condition of
critical sampling critical sampling so
we are having the condition of critical
sampling and we are using practical
low-pass filter
to recover this signal so let's try to
understand if we can recover the signal
in this condition or not in the previous
lecture we saw that the frequency
response H Omega of a practical low-pass
filter will look like this this
frequency here is equal to Omega C which
is the cutoff frequency this frequency
here is equal to minus Omega C and from
minus Omega C to Omega C we call this
pass band we call this pass band and we
are calling it pass band
because when you apply a signal to a
low-pass filter then this signal who
will be reflected at the output only
with the portion between minus Omega C 2
plus Omega C this means the portion of
the input signal between these two
frequencies is passed by the low-pass
filter and therefore we call this pass
band and now we will talk about this top
band stop band is the band of
frequencies for which the low pass
filter we are using will stop the input
signal and it is stopping the input
signal because you can see that the
frequency response of the low pass
filter we are having is equal to zero
when Omega is greater than Omega C so we
call this band of frequencies stop band
and we are calling it stop band because
corresponding to these band of
frequencies the input signal will not
appear at the output that is it is
stopped by the low-pass filter and we
know we use low pass filter to recover
the message signal we obtain the sampled
signal St after performing the sampling
and then we feed this sample signal to a
low-pass filter in our case we are
having the ideal low-pass filter we are
having the ideal low-pass filter and the
output of the low pass filter we call as
recovered message signal so we
represented by M R T now this recovered
message signal may or may not be equal
to the original message signal there are
some conditions which we have discussed
in great detail in the previous lecture
and we know the Fourier transform of the
input signal s T is has Omega and here
you can see the Fourier transform and
the low pass filter we are having which
is ideal in nature is having the
frequency response like this so the
ideal low-pass filter we are having
he is having the frequency response like
this and we have the knowledge of s
Omega and H Omega this means we have the
knowledge of input signals Fourier
transform and the systems frequency
response so if we multiply them we will
get the output signals Fourier transform
mr Omega and if we have Hammar Omega we
can perform the inverse Fourier
transform to get MRT now this MRT may or
may not be equal to the message signal
it depends on various factors now let's
try to understand whether we can have
the recovered signal same as the message
signal when there is critical sampling
along with the ideal low-pass filter
used so we will multiply H Omega and has
Omega and to give you the clear picture
of what actually is happening I will
copy this portion of the waveform and
then I will paste it and finally I will
try to overlay it properly here and now
you can see that according to the
property of low pass filter this portion
of the waveform of s Omega will be
passed and this portion and this portion
will be stopped by the low-pass filter
and therefore M high Omega will have the
waveform like this which is the spectrum
of the message signal the Fourier
transform of the message signal is
having the waveform like this as we have
seen in the first lecture of the
sampling theorem therefore here we can
see that when Omega C is equal to Omega
M we know this frequency here is equal
to Omega M this is twice of Omega M so
here you can see that Omega C the
critical frequency is equal to Omega M
and when this happens we are easily
getting our signal back because M R
Omega we are getting is same as M Omega
and therefore when we recover the
message signal
it will be same as the original message
signal MT so I hope this case is clear
to you that what we are required to do
when there is ideal low-pass filter in
the critical sampling case we are
required to keep our critical frequency
equal to maximum frequency component of
the message signal and when this happens
we can easily extract our signal back
but the question is asking about the
practical low-pass filter it is not
asking about the ideal low-pass filter
so let's move on to the question now we
will understand what will happen when
there is practical low-pass filter in
the case of critical sampling and forced
we will have a look at the frequency
response of the practical low-pass
filter the frequency response H Omega
will look like this and here you can see
that the transition from pass band to
stop band is not this sharp it is taking
some time and therefore here in this
case we are having one more band which
is known as transition band from here to
here we know it is pass band and from
here to here this portion is known as
transition band so there is transition
band and because of transition band this
top band will start from here so this is
our stop band and now we will follow the
same process we will feed our sampled
signal to the practical low-pass filter
and then we will try to obtain M R Omega
and from M R Omega we will obtain M R T
so let's quickly copy this portion of
the waveform and we will try to overlaid
like we have done in the previous case
and now you can see that now you can
clearly see that this portion of the
waveform is passed which is the spectrum
of the message signal
and we need only this portion but along
with this portion this portion and this
portion of these two spectrums are also
passed by the practical low-pass filter
and therefore M R Omega will not be
equal to M Omega and therefore the
obtained or recovered message signal
will not be equal to the message signal
so whenever you have the critical
sampling case in which Omega s is equal
to twice of Omega M and you are using a
practical low-pass filter then the
recovery is not possible so this is the
answer of the question the answer is
false
we cannot recover the message signal
when Omega s is equal to twice of Omega
M with a practical low-pass filter so
option B is the correct option which is
false
now there is one homework problem and in
this homework problem you need to tell
me with a practical low-pass filter
which sampling mode should we use option
is critical sampling option B's under
sampling and option C's over sampling so
try to answer this question and once you
have your answer post it in comment
section and if there is any condition
along with your answer then also mention
that condition so this is all for this
lecture see you in the next one
[Applause]
[Music]
5.0 / 5 (0 votes)