Frequency Domain and Fourier Transforms

1 Chapter 4 Frequency Domain andFourier TransformsFrequency Domain analysis and Fourier Transforms are a cornerstone of signaland system analysis. These ideas are also one of the conceptual pillars withinelectrical engineering. Among all of the mathematical tools utilized in electricalengineering, Frequency Domain analysis is arguably the most far-reaching. Infact, these ideas are so important that they are widely used in many fields notjust in electrical engineering, but in practically all branches of engineering andscience, and several areas of Frequency Content: Combining SinusoidsThe most common and familiar example of Frequency content in signals is prob-ably audio signals, and music in particular. We are all familiar with high musical notes and low musical notes. The high notes do in fact have higherfrequency content than the low notes, but what exactly does this mean?

2 The place to start to answer this question is to consider sinusoids. Recallthat the general expression for a sinusoid at Frequency (or frequencyfinHertz) isx(t) =asin( t+ ) =asin(2 ft+ )When considered as an audio signal,x(t) indicates the changes in air pressureon our ears as a function of time. What is important here is the time variationof the air pressure from some ambient value rather than the ambient value ofthe pressure itself. A negative value refers to that amount below the baseline(ambient) pressure, while a positive amount refers to a pressure higher than thebaseline. c 2000, 2001, 2002 by Sanjeev R. Kulkarni. All rights reserved. Lecture Notes for ELE201 Introduction to Electrical Signals and Systems. Thanks to Sean McCormick and Richard Radke for producing the 4. Frequency Domain AND Fourier TRANSFORMSSo,x(t) being a sinusoid means that the air pressure on our ears varies pe-riodically about some ambient pressure in a manner indicated by the sound we hear in this case is called a pure tone.

3 Pure tones often soundartificial (or electronic) rather than musical. The Frequency of the sinusoid de-termines the pitch of the tone, while the amplitude determines the loudness .It turns out that the phase of the sinusoid does not affect our perception of thetone, which may not be surprising for a pure tone, but is somewhat surprisingwhen we start combining can combine two sinusoids by adding the signals in the usual way. Forexample,x(t) = sin(2 t) + sin(4 t)is a combination of a sinusoid with Frequency 1 Hz and a sinusoid with frequency2 Hz. Here the amplitude of each sinusoid is 1 and the phase of each is 0. A plotofx(t) is shown in Figure The sound created byx(t) is the combinationof the two pure tones that makex(t). Unfortunately, as we ll discuss in moredetail in Chapter XX, humans can t hear the pure tones that comprise the signalx(t) above since the frequencies are too , we can make a similar combination with signals at frequencieshumans can hear.

4 For example, consider the signald(t) = sin(2 350 t) + sin(2 440 t)Each of the two sinusoids (at frequencies 350 Hz and 440 Hz) alone correspondsto a pure tone that can be heard by the normal human ear. Their combination, , the signald(t), makes a very familiar sound, namely the dial tone on astandard telephone line. A plot ofd(t) is shown in Figure Note thatin this figure only 2 hundredths of a second are shown. Because the frequenciesare high, if we showed even a whole second, the signal would oscillate so manytimes (350 and 440 for the constituent sinusoids) that not much useful detailwould be 1 (2*pi*t) 1 (4*pi*t) 2 1012x(t)tFigure : Combining the dial tone is a simple example of a sound that still soundsartificial, by combining more sinusoids at different frequencies we can get SOME MOTIVATING 2 1 : Graph of a dial sounds.

5 Musical notes that we find pleasing largely consist of pure tonesnear the pitch of the musical note, but also contain other frequencies that giveeach instrument its particular qualities. Voice and other natural sounds are alsocomprised of a number of pure , all sounds can be built up out of pure tones, and likewise all timesignals can be constructed by combining sinusoids. Similarly, starting with ageneral time signal, one can break this signal down into its constituent to do this and the consequences of such constructions/decompositions isthe subject of Frequency Domain analysis and Fourier Transforms . First, webriefly discuss two other different motivating Some Motivating ExamplesHierarchical Image RepresentationIf you have spent any time on the internet, at some point you have probablyexperienced delays in downloading web pages.

6 This is due to various factorsincluding traffic on the network and the amount of data on the page , as we will see in Chapter XX, require a substantial amount of data, andas a result, downloading images can be we are browsing through a large database/archive of images andwish to find a collection of images of a particular type. Downloading eachimage completely and then deciding that the bulk of the images are not whatwe are after can be time-consuming. Of course, we may be able to make thisdetermination ( , whether or not the image suits our purposes) with muchless quality than the full image possesses. Certainly we may be able to rejectmost of the images with only a very rough idea of their content. Wouldn t it benice to be able to make this decision with only 1/10 of the data? That way wecould download and rifle through the database much faster to find what we problem is that if we take the standard representation of images (8 bitgray level for each pixel) and send the first 1/10 of the pixels, we will simplyget a portion (namely, 1/10) of the original image, albeit at high 4.

7 Frequency Domain AND Fourier TRANSFORMSOn the other hand if we had a way to represent the coarse (or low Frequency )information separately from the fine (or high Frequency ) information, we couldrequest the coarse information first and only request the additional detail shows an example of an original image, together with xx% of thepixels in the usual format and xx% of the information at the lowest many purposes, the low Frequency version may be adequate to make deci-sions, and it certainly seems more valuable than if we had only the first xx%of the pixels. This notion of a hierarchical representation can be formalizedusing ideas from Frequency Domain turns out that Frequency Domain ideas can help with this browsing problemin a different way as well. In addition to creating hierarchical representationsthat allow sending the most important information first, one might considerreducing the total amount of data in the first place.

8 As we will see, one of thestandard methods for image compression known as JPEG is based on frequencydomain ideas. This is the subject of data compression, which will be discussedin Chapters XX and and TV TransmissionRadio, television, and some other forms of communication ( , cell phones)transmit information via electromagnetic waves. The various sources in theseapplications can be transmitting simultaneously and in the same geographicregion. But how is it that we can tune in to a specific radio station, televisionprogram, or individual with whom we re communicating, rather than hearingthe jumble of all the various transmissions put together?The answer is that different transmissions agree to use different , even though all the signalsare jumbled together in the time Domain ,they are distinct in the Frequency Domain . With some basic Frequency domainprocessing, it is straightforward to separate the signals and tune in to thefrequency we re interested A Trivial Frequency DecompositionBefore discussing Frequency representations for general signals, we consider anexample that is trivial but is still somewhat illustrative.

9 Consider a situation inwhich we are interested in the values of a signalx[0] andx[1] at only two timesn= 0 andn= we have access to the two values of the signal, and wish to con-vey these values to a friend. We could simply convey the valuesx[0] andx[1]themselves and we re , consider the the following alternative scheme. Define a new signalX[0] andX[1] byX[0] =x[0] +x[1]X[1] =x[0] x[1] A TRIVIAL Frequency DECOMPOSITION5(a)(b)(c)(d)Figure : Sending image data in usual format vs. sending only low frequencydata. a) 25% of data in usual format. b) of data in usual format. c) 25%of lowest Frequency data. d) of lowest Frequency 4. Frequency Domain AND Fourier TRANSFORMSWe can then convey the valuesX[0] andX[1] to our friend, and using thesevalues the friend can recover the original signalx[0] andx[1]. Namely, givenX[0] andX[1], we getx[0] andx[1] byx[0] =12(X[0] +X[1])x[1] =12(X[0] X[1])This example seems trivial, but even this simple case illustrates some inter-esting properties.

10 For example, in the new scheme, information onbothvaluesofx[0] andx[1] are being sent simultaneously partially inX[0] and partiallyinX[1]. If we receive onlyX[0] then we don t knoweitherof the original signalvalues, but we know something about ,X[0] is the low Frequency content of the signalx[0], x[1], This isthe part of the signal that doesn t change, or that s common to bothx[0] andx[1]. It is a sort of an average betweenx[0] andx[1] (in fact, it s exactly twicethe average).X[1] is the high Frequency content of the signal the part thatchanges betweenx[0] andx[1]. In fact, the equation shows that it is simply thedifference betweenx[0] andx[1].We can think of the signalX[ ] as just a different representation for the signalx[ ] since we can easily go back and forth between the two representations (usingthe equations above).