Question Topics :
All Questions
Answer: You must have been, in some of the lectures, that 2-D wavelets provide information on vertical, horizontal and diagonal features of images. In fact, one could verify this by taking the 2-D wavelet transform of an image - particularly an image which has clearly distinguishable vertically significant, horizontally significant and/or diagonally significant features. The 2-D wavelet transform involves a 1-D wavelet transform on each of the dimensions. Beyond horizontal, vertical and diagonal features, one would need to see if a feature manifests itself in one, or more than one, band. A 'feature' could mean a particular edge pattern, or a shape for example. One would then have to look for that manifestation in a test image. This is a preliminary answer. I guess if I understand your specific context better, a more pointed answer can be given.
Answer: Let us take your last question first. The importance of phase in signal processing is often like a 'necessary evil' - like friction in mechanical systems. Phase is normally required on account of causality. If causality were not required, then one could have real and even Fourier Transforms for real systems (systems with real impulse responses). Thus the phase would be either 0 or 180 degrees. For a real causal system, the magnitude and phase can be related. In fact, one can see this from the fact that for a real causal system, the real and even part of the impulse response can be constructed from the original impulse response, which corresponds to the real component of the spectrum. So too, a similar process can be followed for the imaginary component. If the magnitude of the spectrum is known, the magnitude together with one of these can give information about the phase.
Answer: 'Linear Phase' or 'Nonlinear phase' is a term which can only be used only with systems which are linear, shift invariant and which admit a frequency response. Linear phase essentially means no dispersion, that is, all sinusoidal components suffer the same time shift and are not RELATIVELY shifted. Nonlinear SYSTEMS cannot strictly be associated with a phase response.
Answer: Notionally, you could take it to be radians per sample. Remember, 'sample' itself is without units. It is accurate to say that the normalized angular frequency indicates the angle covered in one sample time. The maximum allowable angle that can be covered in one sample time is therefore 'pi' radians, giving exactly two samples per cycle.
Answer: There are several researchers who have investigated the applications of wavelets in power system transients. There are two immediate domains which I can think of - in analyzing transient patterns and in analyzing the period-to-period variation of waveforms that are supposed to be periodic. If I understand correctly (I am not an expert in power systems), power system protection will require the analysis of transient patterns to detect possible faults. Wavelets have been used successfully there.
Answer: Good question. Length normally refers to the length of the impulse response. For FIR filters, order = length - 1. For IIR filters, the length is always infinite! However, order in a more general case refers to the number of un-cancelled poles in an IIR filter. The terms 'numerator order' and 'denominator order' are also sometimes used to distinguish these two self-explanatory ideas. To be precise, the 'element cost' of implementation of a rational filter is measured by its 'numerator order' + its 'denominator order'. A 'zero order' filter will simply be an amplifier. Its frequency response will be of constant magnitude.
Answer: The unit step is discontinuous at t=0. Its value at that point is undefined. That does not affect its behaviour under the integral sign, though.
Answer: One can, in principle find out the impulse response of almost any system. However, only for linear time-invariant systems, can one use the impulse response to characterize the system completely.
Answer: You could think of it as approximated by what happens, when many sine waves with frequencies close to one another come together, all reaching their maximum at time = 0. The other way is to think of it as a very narrow pulse through a gating circuit, constructed out of op-amps.
Answer: Getting an impulse in the response to a bounded input indicates instability, since the output is then not really bounded.
Answer: Certainly - look at mobile phones with advanced features, look at MPEG players !
Answer: The Fourier series is meaningful only for periodic signals. The Fourier Transform could be viewed as a limiting case of the Fourier series with the fundamental frequency tending to zero or the period tending to infinity.
Answer: I suggest you read the text on Digital Signal Processing authored by Johnny R Johnson.
Answer: Dear Sibghatullah, The book by Oppenheim, Willsky and Young is fairly basic. The book by Lathi is good from the point of view of excellent illustrations from real life systems and down-to-earth explanations. There is actually a book by Nagrath and some others from BITS Pilani which I also liked, because they had given good numerical examples and practical illustrations.
Answer: Dear Manoj, I appreciate this feedback. It is very valuable to me. It is because of enthusiastic and committed viewers like you that I feel satisfied with this effort and feel like doing more. Thank you once again. With warm regards. Gadre.
Answer: A little outside the scope of this course. Anyway, Statistical variance: the expected value of the squared deviation of a random variable from its mean. Statistical Co-variance of two random variables: the expected value of the product of the deviations of the two random variables from their respective means Please look up any standard text on probability and statistics for more details
Answer: The best example demonstrating instability of the differentiator is the bounded, analytic input: x(t) = sin(a t^2) that is, the sine of any multiple ('a' times) the square of time t. It is clearly bounded by 1. However, its output is: 2at cos (a t^2)which is clearly unbounded. This clearly indicates instability with a reasonable input. However, one could use the square wave, but the square wave is not differentiable in the sense of functions. We have to take recourse to the generalized function, the impulse which is of course, 'unbounded' if you wish to call it that.
Answer: Dear Anindya, For a periodic signal, we are dealing with the restricted space of such signals confined to a period, to make them square integrable/ absolutely integrable and so on. All the inner products, norms should then be calculated in this restricted period and not from (minus infinity).
Answer: Dear Saleem, We (a colleague from Pune and I) are writing a book that will cover this material. We shall keep all of you informed on the progress of the book. Thanks for your interest.
Answer: Dear Manoj, Power spectral density and energy spectral density are respectively relevant for signals for finite power but infinite energy; and signals with finite energy. Parseval's theorem needs to be interpreted accordingly. It is the modulus SQUARED of the FFT which indicates PSD/ ESD.
Answer: Dear Manoj, If the number of FFT points is N, then the spacing between the points on the actual frequency axis, is 'sampling frequency divided by N'.
Answer: It is a little involved. This is essentially a linear Frequency modulated function. Look up the calculation of the spectrum of FM waves in a standard text on communication. Normally these explain the calculation for a single tone modulation, but here you have a linear FM. In a single tone modulation, you would have to use Bessel functions. You could also try to 'complete the square' in the phase term of the Fourier integral, after decomposing cos(..) into a sum of two exponentials and proceed therewith. Please note that the Fourier transform would not be a function in the standard sense, generalized functions would have to be employed.
Answer: Typically, the intersection of the bands of the two signals, since the Fourier Transforms get multiplied. Note that 'bandwidth' may not be defined for one or both the signals.
Answer: The Fourier Transform can be obtained from the Laplace Transform only if the Fourier Transform exists and is analytic. Take the standard ideal filters, the Fourier Transform of the impulse response (frequency response) is not analytic, it has a discontinuity. In that case, we could perhaps make a continuation to the entire complex plane, to get a Laplace transform but that is not useful at all. The Fourier Transform gives important sinusoidal component insights, indicates spectral requirements and processing needed. The Laplace Transform indicates possibilities for realization. Each has its own importance.
Answer: The real axis of the Laplace variable gives the decaying/ growing exponential time term. The imaginary axis gives the oscillatory, sinusoidal term associated with or multiplying the exponential.
Answer: A sinusoid, cos(wt) comprises of a 'positively rotating' phasor exp(jwt) combined with a 'negatively rotating' phasor exp(-jwt). Only when these come together, is a sinusoid formed. It is mathematically more convenient to deal with phasors than sinusoids directly. Thus 'positive' and 'negative' frequencies refer to the frequencies of the phasors, not the sinusoids. 'Negative' frequency is as real or imaginary as 'positive' frequency in this sense.
Answer: Look up standard textbooks recommended for study in the Graduate Aptitude Test of Engineering (GATE) by the IITs. That could be an indicator.
Answer: An analytic function has continuous derivatives of all orders 'almost everywhere', to speak informally. It refers to a function which is just about as smooth as it can get - examples are sinusoids, exponentials.
Answer: There was a whole special issue of the IEEE Transactions on Medical Imaging, circa 2005 or so, devoted to the Applications of Wavelets in MRI! There was a similar special issue of the IEEE Journal on Selected Areas in Biomedical Engg (or some such similar name) around the late 1990s devoted to wavelets/ time-frequency methods in Biomedical Engineering. Even in special issues of the IEEE Proceedings (April 1996), IEEE Trans on Information Theory (March 1992) there have been papers devoted to the application of wavelets in MRI/ Biomedical applications.
Answer: Dear Kiran, 'Audio' is much more than speech -it could include music, percussion and almost anything systematically generated in the audible range by humans. Speech is human communication using an underlying language, with focused intent. Voice is the medium in the human body for speech.
Answer: Dear Arun, DTFT and DFT work with normalized frequencies, so 'radians' or Hz/Hz can be thought of as 'units'. They are not really units, as the frequency is normalized. For the Fourier series, the unit is the same as the original signal as the expansion is in the same domain. For the Fourier Transform, the axis corresponds to frequency (Hz) or angular frequency (radians/sec).
Answer: See my answer to the question by Arun Kumar on 21-10-2012. The Fourier Transform gives insights, which the Laplace Transform sometimes cannot. For example, look at the ideal filters - lowpass, bandpass, and so on. The Fourier Transform immediately tells us what the sinusoidal response should be, which is what the filters are supposed to do, in the first place. The Laplace Transforms of the impulse responses can probably be evaluated in some complicated way to yield non-analytic functions. However, they would not give meaningful or easily understandable interpretations.
Answer: Look up a text on 'Network Synthesis', example, the one by Van Valkenburg. There are several approaches to synthesize positive real functions based on L,C,R elements.
Answer: The inverse z-transform is required for going back to time from the z-domain. All signals cannot be expressed in the frequency domain, exponentially growing signals are a counter-example.
Answer: It is essentially a difference in perspective. The DFT corresponds to a finite length signal. However, the inverse DFT expression, treated as a function of the time index, is periodic in the time index and is hence termed a DFS (Discrete Fourier Series). This is a manifestation of time domain aliasing.
Answer: The DCT (Discrete Cosine Transform) is a small modification of the DFT where only real basis sequences, constructed out of sinusoidal sequences, are used. It is much more amenable to efficient implementation, when real data processing is a must.
Answer: Essentially the real data processing convenience.
Answer: I do not understand the question properly. On what basis are you 'resampling'? Please clarify or reframe the question in more detail.
Answer: Good to know that the lectures are helpful to you. Prof Aditya Abhyankar and I are trying to put a book together based on the series of these lectures. We could send you a very raw form of the book if it helps. You could give us feedback on the content and suggest how it should evolve. This is a general offer to all readers/ audience of our lectures series on this site. You will need to mention your email and give a brief background of yourself to enable us to know who is making the request. You can write to me on the email addresses: vmgadre@ee.iitb.ac.in, vmgadre2012@gmail.com
Answer: same as your previous question - already answered
Answer: Correlation of x(t) with y(t) is akin to convolution of x(t) with y(-t). Autocorrelation of x(t) = Correlation of x(t) with x(t). The autocorrelation indicates the 'similarity' of a signal with its own translates. Cross-correlation of two signals indicates their mutual 'similarity' at different translations.
Answer: If one writes a Fourier series for a periodic square wave and seeks the limit of the series at the point of discontinuity, the limit is shown to converge to the average of the two values across the discontinuity. One could extend this to the Fourier Transform in the generalized sense for the unit step and ask whether its inverse yields a limiting value at the discontinuity. That is probably the line of thought. The other way to understand is to treat the unit step as comprising of a 'DC' part and a 'zero average AC' part - the 'DC part' is the average = 1/2, the 'zero average' AC part takes the value 1/2 for t>0 and (-1/2) for t<0.
Answer: 1. Certainly. In fact, you could look up a paper in the March 1992 issue of IEEE Trans Information Theory on Optimal Choice of wavelets for signal representation. 2. The length, in case of a finite length sequence, puts a constraint. Every time you decompose, you are halving the length of individual sub-band components.
Answer: Question blank
Answer: Not relevant to this forum.
Answer: DFS and DFT are relevant only for 'power' sequences, their energy would diverge.
Answer: That is not true. One could analyze it easily using analog techniques at times. What Digital Processing offers in general, is flexibility, versatility and easy upgrade/ metamorphosis
Answer: Normalized frequency is the ratio of the actual frequency to the sampling frequency. It is used to build Discrete System Theory independent of the sampling rate.
Answer: Typically, probability 'distribution' function is often used to denote the cumulative distribution, which is the running integral of the probability density function from minus infinity to the argument value at which it is being calculated. At any given value of the random variable, the cumulative distribution refers to the probability that the random variable could be less than that particular value.
Answer: I am not clear why you feel that the coefficients are almost zero. Is it because you think that the integral is divided by T, which tends to infinity? But then that is again annulled by the fact that you are bringing impulses arbitrarily close together - and the division by T is absorbed in the 'impulses merging', so to speak. The Fourier Transform can be non-zero over a significantly large region of support in general. Maybe you could pose your question more specifically/ differently after reading this answer.
Answer: One way is indeed to look for a wavelet that is most 'similar' to the signal, which can be determined by looking at the cross correlation with a normalized signal. That would often give the most compact representation. However, that may not be the only consideration. One may want to find the best wavelet to determine specific features that can help classify or characterize the signal/ image. In that case the considerations for choice of wavelet will be different.
Answer: One can broadly determine the level upto which a signal can be decomposed depending upon two considerations: one, how much data one has to decompose as the data gets halved at each decomposition; and second, the extent to which one wishes to get frequency resolution at lower frequencies. The Fast wavelet transform is an efficient way to IMPLEMENT the discrete wavelet transform. It is not a different transform.
Answer: Thank you very much for these encouraging words. Believe me, it is the encouraging response from you and everyone else on this forum that makes me want to answer queries as regularly as I can. It makes me very happy to see many people in different places using these lectures and this forum. God bless you.
Answer: Certainly but you could also probably do that without listening to the lectures. Please listen to the lectures for much more, that just this little objective. Else you will miss much of what is being explained, on account of a fixation on a very narrow objective!
Answer: Not sure what you mean by the 0 to pi spectrum. I think what you are referring to, is a 'prototype' spectrum which has distinct amplitudes at each frequency between 0 and pi. A simplest example of such a spectrum is one whose magnitude varies linearly from 0 to pi. I have used that to illustrate many filter bank operations. Vanishing moments refer to the degree of the polynomial which can be 'annihilated' (reduced to zero) at the output of the highpass filter of the analysis filter bank. I suggest you take any text/ reference which discusses how the Daubechies' filters are constructed. They have an increasing number of vanishing moments with an increase in order. I have also explained these in my NPTEL lectures where the Daubechies' filters are derived.
Answer: Yes, the CWT is the natural choice if you wish to see the cross-correlation of the signal with a wavelet at different scales and translates. The scaling and translation parameters are continuous in the CWT. To get an idea of the scale range to work with, you will need some a-priori knowledge of the signal spectrum. Remember, a particular scale parameter value corresponds to a certain range of frequencies, based on the bandpass nature of the wavelet.
Answer: Yes, indeed, a sequence defined only at discrete, isolated instants is a discrete-time sequence even if the points at which it is defined do not coincide with the integers. In the particular example which you give, the sequence is still uniformly spaced with a spacing of 0.25 between samples. One could also conceive of a non-uniformly sampled discrete time sequence. A theory to deal with that kind of sequence, would be more difficult than standard discrete system theory.
Answer: The 's' and 'z' domains are parameter domains for complex continuous time and discrete time exponentials respectively. Yes, realistic linear shift-invariant systems exhibit decaying exponentials for their natural responses. It becomes meaningful to characterize systems in terms of these exponentials, and therefore the 's' and 'z' planes.
Answer: The DFT is, in a way, a discretization or sampling of the DTFT, for a sequence of finite length. If a sequence be of finite length, then the degrees of freedom are the points at which the sequence is supported. Therefore the frequency domain needs to reflect that finite number of degrees of freedom. Therefore it is adequate to sample the DTFT at as many points as the length of the support of the sequence. Here again, you could explain your question or repeat if your specific point has not been addressed.
Answer: Actually some more lectures have already been recorded. I shall request NPTEL to put them up on the site if many are interested. Further, we propose to keep building on the course contents through more lectures and supplementary course material. We would also welcome suggestions from viewers and users.
Answer: 1. About the inverse upsampling operation: The upsampling operation is invertible, by simply removing the (M-1) zeros that one inserts between the successive original samples. In the z-domain, this means replacing z by (z^(1/M)). 2. As far as identifying the original samples is concerned (what you are probably calling the 'non-zero element'), the machine could simply take every Mth sample - driven by a counter/clock. This is easily possible because the location of the desired samples is known a-priori.
Answer: There is a close relationship between the seemingly different 'kinds' of uncertainty. Think of time as analogous to one-dimensional space. Then 'time uncertainty' is clearly like 'positional uncertainty'. The other analogy is more subtle. Remember the derivation for uncertainty that relates the frequency domain uncertainty to the energy in the derivative with respect to time. If we use that result, then we are essentially talking about an energy pertaining to the derivative with respect to one-dimensional space, which is analogous to velocity. That is, the change in location is analogous to the velocity.
Answer: Dear Santhana, I have a feeling that I have made a mistake when saying the values of L1 and L2. I probably intended to say something different. Could you please tell me in more detail and suggest what should have been the correct thing that you expected? I shall also check myself then. Thanks for pointing out Best wishes Gadre
Answer: Dear Kamlesh, You could apply the JPEG2000 filters SEPARABLY in the three directions - to give the coefficients. That would give eight bands in each level of decomposition.
Answer: Should be easy to do. The filters are just two-sample long with integer coefficients and one could work out an economical realization with the lattice structure (Polyphase components). See one of my lectures explaining this. That should be easy to realize on a DSP.
Answer: psi(t) and ALL its dyadic dilates and integer translates TOGETHER span the space of square integrable functions L2(R). phi(t), with its specific dilate at a particular power of two, and integer translates of this dilate span the subspace of L2(R) giving information UPTO a particular resolution. The corresponding dyadic dilate of psi(t) and the integer translates give INCREMENTAL information in the immediately next resolution. Both psi and phi play their role in capturing details, phi, UPTO a particular resolution and psi, the incremental detail AT a particular resolution.
