Why we need Frequency and Time information at the same time

The big disadvantage of a DFT is that it has only frequency resolution and no time resolution. This means that although we might be able to determine all the frequencies present in a signal, we do not know when they are present. So we go for Wavelets.

My question is, What are the applications (other than music & speech processing ) that require both the time and frequency information at same time ?

How do you relate that we require both spatial (time) and frequency information in an image at a same time ?

Give me some good examples and suggest a book (PDF links) to learn Wavelet Multiresolution Analysis (For Dummies)

• I think that it would be easier to list the applications that do not require time information. However, your statement that the DFT only has frequncy information is inaccurate. If the DFT discarded the time information, then the inverse DFT would not be possible. – nispio Feb 20 '14 at 23:41
• Yes, I agree.. I didn't say the time information will be discarded. I said that we could only see the frequency information and we do not know when they are present. So – Premnath D Feb 21 '14 at 15:12

For the source, go to end of the answer


Suppose one day you got one note which has some thing written to it, say "Major frequency components are 10 Hz, 25Hz, 50 Hz and 100 Hz". Somehow, you understood that its time-series representation is a very important thing (may be master-piece work of a great musician, or some national security matter, anything). So you decided to find it.

You start by plotting the spectrum which looks like below:

(There can be some noises/disturbances in between these frequencies, but most people neglect it)

Now you want to find its time-series representation. Since we have worked it a lot in our signal processing class in college, most probably, we get below signal:

$$x(t)=cos(2 \pi 10 t)+cos(2\pi 25 t)+cos(2 \pi 50 t)+cos(2 \pi 100 t)$$

Great !!!

But is this the only time series representation for those frequencies?

Consider another time series signal which plays a sine wave of frequency 100 Hz for 300 ms, then 50 Hz sine wave for next 300 ms, then 25 Hz sine wave for 200 ms, then 10 Hz sine wave for 200 ms. A total of 1000 ms long signal. If you plot it, it will look like below:

Actually above signal also have similar frequency spectrum (shown below):

Those ripples are because of sudden changes in frequency. Amplitude changes are because of change in how long each frequency was played. They are not of importance to us right now.

So what to do now? How will you pick one? Even if we change the order they were played, still we get the same frequency spectrum. So how many such combinations you have?

In short words, without knowing when those frequencies were played, it is very difficult to find its time series representation, and that is where, wavelets come into picture. It provides not only the frequencies present in the signal, but also, the time it was present in the signal.

UPDATE (17-Feb-14)

Question : How do you relate that we require both spatial (time) and frequency information in an image at a same time ?

Answer : Me too, had the same question before, so after seeing your comment, I thought I should find one example (I haven't studied wavelets in the context of images before wavelets). And below explanation is the summary of an answer in Quora.com (all credits to that author, Ron Reiter) which I felt as a good example.

You might be knowing that we use the DCT for jpeg compression of images. It preserves the low frequency data while throws away some of the high frequency data in the images, based on the image quality we need.Or in other words, this method is good with low frequency data, but not that accurate with high frequency data.

For example, in an image with sky and a tree, the areas of sky will be preserved accurately while some details of tree will be thrown off.

In DCT, we take 8x8 blocks of images and encode it for all frequency of data. For very low frequencies, we could have used higher block size while for very high frequencies, we may have to use lower block size than this.

Now apply our wavelet method as explained in 1D case above. What if we can find both frequency and their location in the image? We can encode the low frequency areas in big chunks while high frequency areas in low chunks. So it produces better results.

SOURCE:

1. THE WAVELET TUTORIAL by ROBI POLIKAR
• All the images, explanation, everything, are extracted from this link. It is a must-read introduction of wavelets to beginners
2. Amara's Wavelet Page
3. Wavelet in image compression - Ron Reiter at Quora.com
• Can you state any other applications? – Premnath D Feb 15 '14 at 18:12
• How do you relate that we require both spatial (time) and frequency information in an image at a same time ? – Premnath D Feb 16 '14 at 3:48
• updated the answer. – Abid Rahman K Feb 17 '14 at 2:48

Yes, the way is from segment wise DFT to overlapping segments with window functions to something like Gabor wavelets to more general wavelet transforms. Only the first variant should be avoided for numerical reasons, the others have uses in their own right.

Applications are for instance if you analyze speech or some piece of music, where not only it is important that some pitch pattern occurred, but also when.

And of course, for broadband WiFi the location of information in a time-frequency structure is the basic working principle.

Multiresolution analysis is not for dummies. That is why, for instance, there is no decent wikipedia article on this topic. You immediately dive into some heavy functional analysis. Wim Sweldens once did a historical survey article on orthogonal wavelets that had all essential information.

Or perhaps it was someone else, however, have a look at Jawerth/Sweldens (1994) An overview of wavelet based MRA

• Thank you .. Good examples (I did know about music and speech processing examples) .. can u elaborate on broadband WiFi example.. I am interested to know about it .. – Premnath D Feb 15 '14 at 18:09
• How do you relate this to an image? – Premnath D Feb 16 '14 at 3:46
• @PremnathD: This would be more difficult, since images usually are not dominated by frequency events. The only exception that comes to mind are textures. There I find the approach via the Gauß-Laplace-pyramid and basis decomposition with strongly localized basis functions more intuitive. – LutzL Feb 16 '14 at 7:03
• @PremnathD: Simple signal coding method transmit one signal a time. QAM transmits a complex number at each instant, ODFM transmits a whole vector of complex numbers per tact interval. To do that, the vector components are encoded via FFT as the coefficients of a trigonomentric polynomial, essentially slicing the used frequency band in densely packed subbands. Each subband then can have its own bitrate, optimizing the total bitrate. This was actually already proposed by Shannon in his 1948 paper on this topic. – LutzL Feb 16 '14 at 7:08