ÐÏà¡±á>þÿ =?þÿÿÿ:;<€€Úz€€à€€Ø~ I ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿì¥Á5@ ø¿þXbjbjÏ2Ï2 +bXX³ÿÿÿÿÿÿˆPPPPPPPàH°H°H°8€°„¶àr„·.0Bç"dçdçdç?è?è?èÍÏÏÏÏÏÏ$öRH"Žó9Pï?è?èïïóPPdçdçÛ,¡ú¡ú¡úïÀPdçPdçÍ¡úïÍ¡ú$¡úÅú©ÌPPdç·pÔ8~åÆH°AñØuÍB0r‚Ö"ölÖ"4d$ˆXPPPP`Ö"Pq\?èð/êb¡ú‘ëìÔ?è?è?èóóàà[ä`dO…úààä` Harmonic Analysis on Zn Tomomi Haynes Math 4495 Dr. Derado December 14, 2005 Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms. Also, the basic waves are called “harmonics”, hence the name “harmonic analysis.” Since the name of the field is taken after the great French mathematician Jean Baptiste Joseph Fourier, one may think that the area of study was developed mainly by him. However, nothing is farther from the case. Fourier may be the prominent figure who is directly responsible for the development of Harmonic Analysis in the last two centuries, but he owes his achievements to the mathematicians before him. When describing his achievement in physics and astronomy, Newton once dedicated his achievement to Galileo and Kepler by noting, “If I have seen further [than certain other men] it is by standing upon the shoulders of giants.” This may well apply to the case of Fourier. There were indeed many contributions made by other mathematicians and scientists whose ideas helped Fourier to formulate his idea of decomposing a function into a sum of sinusoids or “waves.” In Trigonometric Delight (1998), Eli Maor eloquently describes the mathematical movement in the 17th and 18th centuries that influenced Fourier’s formulation of the theorem. One of the movements that influenced Fourier’s theorem is the shift in the treatment of trigonometry. Trigonometry had begun to assume its modern and analytic character with the great French mathematician, Francois Viete (1540-1603). With Viete, trigonometry underwent important changes, one of which is admitting infinite processes. In 1593, he discovered the famous infinite product EMBED Equation.3 It was the first time an infinite process was explicitly written as a mathematical formula, and this was the beginning of modern analysis. In England, substantial contributions to trigonometry were also made in the first half of the seventeenth century. One of contributions which had an impact on Fourier’s theorem may be John Wallis’s (1616-1703) work on infinite series that is an immediate forerunner to Newton’s discoveries in the same field. Wallis, more than anyone else up to his time, realized the importance of analytic methods in mathematics: he was the first to treat conic sections as quadratic equations rather than geometric objects. His most famous formula is the infinite product EMBED Equation.3 which together with Viete’s product are regarded as the most beautiful formulas in mathematics. There was another reason for the rise of analytic trigonometry in the first half of the seventeenth century. Instead of applying trigonometry to heavens (spherical problems), mathematicians and scientists were interested in the mechanical world of daily life. The science of artillery was also developed during the century. Another branch of mechanics studied in the seventeenth and eighteenth centuries was oscillations. Due to the great sea voyages of the era, ever more accurate clocks of ever greater precision were heavily in demand. This led scientists to study the oscillations of pendulums and springs of various kinds. On another level, the increased skill and sophistication in building musical instruments motivated scientists to study the vibrations of sound-producing bodies such as strings, membranes, bells and air pipes. What is so significant about all these developments is that they all underscored the role of trigonometry in describing periodic phenomena which resulted in trigonometric functions---one of the essences of Fourier’s theorem. It was not until Euler that trigonometry was fully incorporated with complex numbers for Fourier to later utilize it in his work. With Euler, the subject became truly analytic. Euler’s formula, EMBED Equation.3 , which appeared in his great work, Analysin infinitorum (1748) expresses a deep relationship between trigonometric functions and the complex exponential function. These developments moved trigonometry ever farther from its original connection with a triangle. The outstanding problem in the second half of the eighteenth century was the vibrating string. Pythagoras, in the sixth century B.C., already discovered some of the laws governing the vibrations of a string. However, it was not until after the time of Newton and Leibniz that a full investigation of the problem was possible. This was due to the absence of partial differential equations. For the vibrating string the relevant equation is EMBED Equation.3 , where EMBED Equation.3 is the displacement from equilibrium of a point at a distance EMBED Equation.3 from one endpoint of the string at time EMBED Equation.3 , and EMBED Equation.3 is a constant that depends on the physical parameters of the string, which is its tension and linear density. This equation, known as the one-dimensional wave equation, excited the best mathematical minds of the time. Euler and D’Alembert expressed their solutions in terms of arbitrary functions representing two waves. On the other hand, Daniel Bernoulli found a solution involving an infinite series of trigonometric functions. These two types of solution to the same problem looked so different, the question arose whether they could be reconciled, and if not, to find out which was the more general. This question was answered by Fourier in his most important work, Theorie analytique de la chaleur (Analytic theory of heat, 1822). He showed that almost any function, when regarded as a periodic function over a given interval, can be represented by a trigonometric series of the form EMBED Equation.3 EMBED Equation.3 where the coefficients EMBED Equation.3 and EMBED Equation.3 can be found from EMBED Equation.3 by computing certain integral. We would like to see what he means by it by observing his study on heat propagation through Eli Maor’s account. Fourier was particularly interested in the manner in which heat flows from a region of high temperature to one of lower temperature. The Law of cooling, previously discovered by Newton governed only the temporal rate of change of temperature, not its spatial rate of change. To deal with this problem, one must use the analytic tools of the continuum, in particular partial differential equations. Fourier showed that to solve such an equation, one must express the initial temperature distribution as a sum of infinitely many sine and cosine terms, which is now called Fourier series. This moment was the beginning of Harmonic Analysis. The most remarkable advancement made in Harmonic Analysis in the last century is probably the invention of Cooley-Tukey fast Fourier transform algorithm known as FFT. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size n = n1n2 in terms of smaller DFTs of sizes n1 and n2 recursively, in order to reduce the computation time to O(n log n) for highly-composite n (called smooth numbers). This algorithm, including its recursive application, was already known around 1805 to Carl Friedrich Gauss who used it to interpolate the trajectories of the asteroids Pallas and Juno, but unfortunetely, his work was not widely recognized. The effect of Cooley and Tukey’s algorithm was particularly significant in the area of signal processing, probably the best-known application of Fourier analysis. We would like to see some basic examples of signal processing at the end of the paper. Applications of the use of Fourier transform are endless, for instance, Radon transform was advanced in the early 1900s by the mathematician from whom it takes its name. Fourier analysis are also found in x-ray crystallography, computerized tomography known as CT scanners, in the field of nuclear magnetic resonance, and in radioastronomy and modern Cosmology. The next section of this paper explores the connections between Fourier’s transform and the notions that are essential to the development of Fourier transform. Harmonic Analysis on Zp I Let G EMBED Equation.3 = {e EMBED Equation.3 , EMBED Equation.3 }. (1) Show that G EMBED Equation.3 is a commutative group. (2) For which p G EMBED Equation.3 is a cyclic group. (3) Find a relation to the modular arithmetic. Before answering the questions above, we would like to revisit the notion of roots of unit and a concept of group introduced in abstract algebra; in particular, we want to review the properties of group EMBED Equation.3 . The first reason to this is that EMBED Equation.3 is the set of roots of unity and the second reason is that the operation of EMBED Equation.3 is twofold. That is, EMBED Equation.3 has a complex multiplication as its binary operation; however, multiplying a complex number means adding exponents of EMBED Equation.3 which ultimately leads us to addition modulo p. EMBED Equation.3 th Roots of Unity n th roots of unity are the solutions to the equation EMBED Equation.3 . As the consequence of the fundamental theorem of algebra, we are guaranteed to have n different nth roots of unity. EMBED Equation.3 (k = 0, 1, 2, 3, …, n-1) Recall an important property of roots of unity: the sum of roots of unity is 0; that is, EMBED Equation.3 B. The properties of group EMBED Equation.3 EMBED Equation.3 is a group under addition modulo p. We will show that it is a group by showing that the following properties hold: Closure Associative laws Identity Inverse (a) Closure: Before proving that EMBED Equation.3 is closed in general term, let us look at the case with p = 3. Let EMBED Equation.3 and EMBED Equation.3 . When EMBED Equation.3 , EMBED Equation.3 . When EMBED Equation.3 , EMBED Equation.3 When EMBED Equation.3 EMBED Equation.3 . When EMBED Equation.3 EMBED Equation.3 . So we can say that EMBED Equation.3 is closed under addition modulo 3. This also implies that any number in EMBED Equation.3 , we can find a number in EMBED Equation.3 that is congruent to that number. Notice also that EMBED Equation.3 is a cyclic group. That is, EMBED Equation.3 and EMBED Equation.3 , i.e., 1 and -1 EMBED Equation.3 are generators of EMBED Equation.3 . Because of this property, no matter what integer we use for EMBED Equation.3 to do the operation, all the results of the operation belong to EMBED Equation.3 . This confirms the fact that EMBED Equation.3 is closed under modulo arithmetic. Let us look at the examples to see what it means by a group being cyclic. By definition, when the operation is addition, EMBED Equation.3 is interpreted as EMBED Equation.3 when EMBED Equation.3 is positive. EMBED Equation.3 is interpreted as EMBED Equation.3 when EMBED Equation.3 is negative. EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 , EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 So basically, if a group is cyclic, any number in Z can be expressed in terms of elements in EMBED Equation.3 since we can always find a number in EMBED Equation.3 that is congruent to that number. In other words, all EMBED Equation.3 belong to one of class representatives, {0}, {1}, {2},…, {p-1}. For example, with our case p=3, every number in Z belongs to either {…,-3p, -2p, -1p, 0, p, 2p, 3p,…}, {…, 1-3p, 1-2p, 1-p, 1, 1+p, 1+2p, 1+3p,…}, or {2-3p, 2-2p, 2-p, 2, 2+p, 2+2p, 2+3p,…}. Therefore, EMBED Equation.3 is closed under modular arithmetic. (b) Associative laws: Associative laws hold under addition modulo p because of the fact that associative laws hold under addition.. Let EMBED Equation.3 . Then EMBED Equation.3 This implies that EMBED Equation.3 EMBED Equation.3 (c) Identity: EMBED Equation.3 is the identity of EMBED Equation.3 , if for any k in EMBED Equation.3 , EMBED Equation.3 . So if EMBED Equation.3 is a group under addition modulo p, there is a unique element EMBED Equation.3 in EMBED Equation.3 such that EMBED Equation.3 for all k in EMBED Equation.3 . Since the operation involves addition, the identity of EMBED Equation.3 is 0 just as the identity of EMBED Equation.3 is 0 under addition. For example, when p = 3. EMBED Equation.3 EMBED Equation.3 since 0 and 3 are congruent. EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 since 4 and 1 are congruent. EMBED Equation.3 EMBED Equation.3 , EMBED Equation.3 EMBED Equation.3 since 2 and 5 are congruent. In general, EMBED Equation.3 = k for all EMBED Equation.3 . (d) Inverse: For each element k in EMBED Equation.3 , there is an element m in EMBED Equation.3 , called the inverse of k, such that EMBED Equation.3 . Let us find the inverse of each element in EMBED Equation.3 when p = 3 and derive the generalization from it. EMBED Equation.3 . So -1 is the inverse of 1. The equivalent relation we are dealing with is addition modulo p, so we have to express -1 in terms of one of class representatives. We do so by applying modular arithmetic. EMBED Equation.3 . Since 2 is congruent to -1, we can replace -1 with 2 which is in EMBED Equation.3 . Applying addition modulo 3, we get, EMBED Equation.3 So 1 and 2 are inverse of each other just as 1 and -1 are. 0 + (-0) = 0 and EMBED Equation.3 . So we can say that 0 is the inverse of itself just as integers such as -3n, 3n are inverses of 0 for any EMBED Equation.3 . So in general, we can say that k and p-k are inverse of each other because k + (p – k) = p EMBED Equation.3 mod p = 0. Now we are ready to prove that G EMBED Equation.3 is a group. First, G EMBED Equation.3 is non-empty because EMBED Equation.3 . The binary operation for the set is the complex multiplication which is ultimately the operation under addition modulo p. Now, we need to show that: (a) G EMBED Equation.3 is closed under complex multiplication. (b) Associative laws hold; (c) Identity exists; (d) And inverse exists. (a) Closure: We are going to look at a few examples before proving the general case. Let us begin with the case when p = 2, that is EMBED Equation.3 EMBED Equation.3 . Let k, EMBED Equation.3 = EMBED Equation.3 . Then EMBED Equation.3 . For G EMBED Equation.3 to be a group, it must be closed under complex multiplication. What it means is that EMBED Equation.3 must be in G EMBED Equation.3 . So it ultimately means that EMBED Equation.3 . From our observation on EMBED Equation.3 , we claim that G EMBED Equation.3 is closed if exponents of each element in G EMBED Equation.3 are closed under modular arithmetic. Let us examine this claim with examples using p = 2, and p = 3. When p=2, G EMBED Equation.3 = EMBED Equation.3 EMBED Equation.3 , for k = 0, 1. Let EMBED Equation.3 and EMBED Equation.3 , then EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 . If EMBED Equation.3 and EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 . If EMBED Equation.3 and EMBED Equation.3 , then EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 . If EMBED Equation.3 and EMBED Equation.3 , then EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 . EMBED Equation.3 is closed under complex multiplication because the operation is ultimately the addition modulo p. When p = 3, G EMBED Equation.3 = EMBED Equation.3 EMBED Equation.3 , for k = 0, 1, 2. Let EMBED Equation.3 and EMBED Equation.3 . Then, EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 . If EMBED Equation.3 and EMBED Equation.3 , then EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 . If EMBED Equation.3 and EMBED Equation.3 , then EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 . If EMBED Equation.3 and EMBED Equation.3 , then EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 . If EMBED Equation.3 and EMBED Equation.3 , then EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 . So again, EMBED Equation.3 is closed under complex multiplication because the operation on exponents is modular arithmetic. Generalizing this, we can say that EMBED Equation.3 is closed under complex multiplication because EMBED Equation.3 for p e" 1 is a group under modular arithmetic. So for any EMBED Equation.3 , EMBED Equation.3 . (b) Associativity: Let EMBED Equation.3 . We need to show that EMBED Equation.3 that is, it is equivalent to showing that EMBED Equation.3 . EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 . Therefore, the associative law holds for EMBED Equation.3 . (c) Identity: Let EMBED Equation.3 be the identity of EMBED Equation.3 . By definition of identity, EMBED Equation.3 is the identity of EMBED Equation.3 if EMBED Equation.3 for any EMBED Equation.3 in EMBED Equation.3 for any EMBED Equation.3 . This implies that we simply need to find some EMBED Equation.3 , such that EMBED Equation.3 for any EMBED Equation.3 for the exponent of EMBED Equation.3 . From the preliminary observation made on EMBED Equation.3 , we know that k must be 0. So adding the inverse of EMBED Equation.3 , which is EMBED Equation.3 , to both sides, we obtain EMBED Equation.3 . Thus, EMBED Equation.3 if EMBED Equation.3 . So 1 is the identity of EMBED Equation.3 , where k = 0 is the identity of EMBED Equation.3 . Inverses: By definition of inverse under multiplication, EMBED Equation.3 is the inverse of EMBED Equation.3 if EMBED Equation.3 * EMBED Equation.3 EMBED Equation.3 = EMBED Equation.3 . This means that we have to find some number EMBED Equation.3 which satisfy the modular arithmetic (k + k EMBED Equation.3 ) mod p = 0. But this has been shown in the preliminary investigation showed that for any k, p-k is the inverse of k. So we know that EMBED Equation.3 and EMBED Equation.3 for any EMBED Equation.3 are the inverse of each other, since EMBED Equation.3 * EMBED Equation.3 = EMBED Equation.3 Let us look at some examples using p = 3. EMBED Equation.3 , for k = 0, 1, 2. The inverse of EMBED Equation.3 is itself since k = 0 and p-k = 3-0 = 3, k-(p-k) = 0 + 3 = 3, and EMBED Equation.3 , so EMBED Equation.3 = 1 = EMBED Equation.3 . The inverse of EMBED Equation.3 is EMBED Equation.3 since k = 1 and p – k = 3 – 1 = 2 and EMBED Equation.3 . So generally speaking, for any EMBED Equation.3 , if EMBED Equation.3 , then EMBED Equation.3 and EMBED Equation.3 are inverse of each other. Finally, we are ready to answer the questions listed at the beginning of this section. (1) Show that EMBED Equation.3 is a commutative group. What needs to be shown here is that EMBED Equation.3 is Abelian. But this is obvious because the binary operation used for EMBED Equation.3 is the multiplication of complex numbers and multiplication is commutative. Under this operation, we need to apply modular arithmetic on exponents of EMBED Equation.3 but this operation is also commutative under addition modulo p. So we have, EMBED Equation.3 . (2) For which p EMBED Equation.3 is a cyclic group. EMBED Equation.3 is a cyclic group if an element in EMBED Equation.3 generates every element in EMBED Equation.3 , that is, EMBED Equation.3 = EMBED Equation.3 . This ultimately means that k must be a number that generates all elements of EMBED Equation.3 when multiplied by EMBED Equation.3 . Let us look at a few examples to see what we can conjecture about the relation between p and k. When p = 2, EMBED Equation.3 for k = 0, 1. When k = 0. EMBED Equation.3 for any EMBED Equation.3 . So, k = 0 fails to be a number for which EMBED Equation.3 is a generator of EMBED Equation.3 . The relation between 0 and 2 are not clear. So we proceed with more observation on EMBED Equation.3 . When k = 1. EMBED Equation.3 . So k = 1 is a number where EMBED Equation.3 generates all elements of EMBED Equation.3 when raised to EMBED Equation.3 power where EMBED Equation.3 . For example, EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 and so forth. So EMBED Equation.3 is a generator of EMBED Equation.3 . It looks like EMBED Equation.3 is the relation we need for generating all elements of EMBED Equation.3 , but what is the relation? We take a look at one more example since this example does not seem to be sufficient to conjecture any thing for the relation between k and p. So let us look at another example with p = 3. When p = 3, EMBED Equation.3 for k = 0, 1, and 2. When k = 0. EMBED Equation.3 for any EMBED Equation.3 . So, k = 0 again fails to be the number for which EMBED Equation.3 is a generator of EMBED Equation.3 . When k=1. EMBED Equation.3 . So k = 1 is again a number where EMBED Equation.3 generates all elements of EMBED Equation.3 . For example, EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 and so on. So it appears that EMBED Equation.3 is the relation between k and p for which EMBED Equation.3 is a generator of EMBED Equation.3 . What about EMBED Equation.3 , where EMBED Equation.3 ? EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 , and EMBED Equation.3 . From these two examples, it appears that when EMBED Equation.3 and EMBED Equation.3 are relatively prime, EMBED Equation.3 generates every element of EMBED Equation.3 . So our conjecture is EMBED Equation.3 is a generator of EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Now we prove that the conjecture is true. Suppose that k and p are not relatively prime. Then there exists a number other than 1 that is the greatest common divisor of EMBED Equation.3 and EMBED Equation.3 . This means that EMBED Equation.3 and EMBED Equation.3 for some EMBED Equation.3 . Then EMBED Equation.3 is a generator of EMBED Equation.3 , call it EMBED Equation.3 . Since EMBED Equation.3 is a generator of EMBED Equation.3 , EMBED Equation.3 . Notice that EMBED Equation.3 . So if we let EMBED Equation.3 EMBED Equation.3 should generate an element of EMBED Equation.3 other than EMBED Equation.3 itself since EMBED Equation.3 . So EMBED Equation.3 which implies that EMBED Equation.3 by theorem. This is true only when EMBED Equation.3 . So EMBED Equation.3 cannot be a generator of EMBED Equation.3 . Suppose that EMBED Equation.3 is not a generator of EMBED Equation.3 . Then there exists j in Z such that EMBED Equation.3 is not a generator of EMBED Equation.3 . Since k and p are relatively prime, there is no common factor other than 1. That means that EMBED Equation.3 generates all elements in EMBED Equation.3 since EMBED Equation.3 for all EMBED Equation.3 . This is a contradiction to our hypothesis that EMBED Equation.3 is not a generator of EMBED Equation.3 . Thus EMBED Equation.3 is a generator of EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Q.E.D. (3) Find a relation to the modular arithmetic. We observed that the operation on EMBED Equation.3 is twofold. Multiplying complex numbers means applying addition modulo p on the set. So if we describe the relation between EMBED Equation.3 and EMBED Equation.3 in terms of function, we can say that EMBED Equation.3 is a homomorphism. A homomorphism is a mapping that preserves the operation of the group that is used as a domain of the function: that is, EMBED Equation.3 : EMBED Equation.3 EMBED Equation.3 is called a homomorphism if EMBED Equation.3 for all EMBED Equation.3 . We need to show that this claim is valid. To do so, we first need to define the function and show that EMBED Equation.3 for all EMBED Equation.3 . EMBED Equation.3 Define EMBED Equation.3 : Z EMBED Equation.3 EMBED Equation.3 by EMBED Equation.3 We show that EMBED Equation.3 is a homomorphism. EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Thus EMBED Equation.3 is a homomorphism of Z EMBED Equation.3 into EMBED Equation.3 . Now we want to see if EMBED Equation.3 is an isomorphism. By definition, EMBED Equation.3 is an isomorphism if it is a bijection. So we first check to see whether EMBED Equation.3 is an injection then check to see if it is a surjection. The function EMBED Equation.3 is injective if EMBED Equation.3 . We prove this by proving the contrapositive statement of the original statement. That is if EMBED Equation.3 then EMBED Equation.3 . Let EMBED Equation.3 and EMBED Equation.3 . Then, EMBED Equation.3 . This means that EMBED Equation.3 and EMBED Equation.3 . So we proved that if EMBED Equation.3 then EMBED Equation.3 . To prove that EMBED Equation.3 is surjective, we need to show that if for any EMBED Equation.3 there exists a EMBED Equation.3 for which EMBED Equation.3 . Let EMBED Equation.3 , then there exists EMBED Equation.3 for which EMBED Equation.3 and EMBED Equation.3 since all integers are congruent to one of the members of EMBED Equation.3 . Furthermore, EMBED Equation.3 for EMBED Equation.3 for the same reason. Thus EMBED Equation.3 is an isomorphism. II Let Z EMBED Equation.3 = {0, 1, 2,…, p-1}. Let H be a set of all functions on EMBED Equation.3 into C. Define EMBED Equation.3 (4) Show that EMBED Equation.3 is an inner (dot) product. To show that EMBED Equation.3 is an inner product, we need to show that the following axioms are satisfied for all EMBED Equation.3 and EMBED Equation.3 in EMBED Equation.3 , and EMBED Equation.3 . (a) EMBED Equation.3 = EMBED Equation.3 (b) EMBED Equation.3 (c) EMBED Equation.3 (d) EMBED Equation.3 and EMBED Equation.3 if and only if EMBED Equation.3 Properties used in proofs are as follow, and we will refer to them by the numbers corresponding to them: For any complex numbers EMBED Equation.3 , EMBED Equation.3 ---------- (1) EMBED Equation.3 ----------------------- (2) EMBED Equation.3 ------------------- (3) EMBED Equation.3 ----------------- (4) Proof of (a) EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 , by (1) EMBED Equation.3 , by (2) EMBED Equation.3 EMBED Equation.3 Proof of (b) EMBED Equation.3 = EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 +…+ EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 = EMBED Equation.3 Proof of (c) EMBED Equation.3 EMBED Equation.3 ) EMBED Equation.3 , by definition. Proof of (d) EMBED Equation.3 = EMBED Equation.3 , by (3) EMBED Equation.3 EMBED Equation.3 if and only if EMBED Equation.3 EMBED Equation.3 =0 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 . Notice: The roots of unity could be understood as functions in H. Precisely, EMBED Equation.3 is defined by EMBED Equation.3 EMBED Equation.3 (5) Show that EMBED Equation.3 are orthogonal i.e., EMBED Equation.3 if EMBED Equation.3 and EMBED Equation.3 . Let us compute EMBED Equation.3 when p = 3. EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Let us compute EMBED Equation.3 . EMBED Equation.3 = EMBED Equation.3 EMBED Equation.3 Let us compute EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Now we would like to compute general case of EMBED Equation.3 . EMBED Equation.3 EMBED Equation.3 If EMBED Equation.3 EMBED Equation.3 If EMBED Equation.3 then EMBED Equation.3 . Since EMBED Equation.3 the exponents of each term is distinct. So, EMBED Equation.3 This implies that we are adding the sum of roots of unity. By the theorem we learned, know that the sum of roots of unity is 0. Thus, EMBED Equation.3 If EMBED Equation.3 , then EMBED Equation.3 . So the sum of all terms would be EMBED Equation.3 because we would have to add 1 for p times. Thus, EMBED Equation.3 are orthogonal, that is, EMBED Equation.3 if EMBED Equation.3 Below is the computation of EMBED Equation.3 which is the case when EMBED Equation.3 . EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 = EMBED Equation.3 EMBED Equation.3 . This implies that EMBED Equation.3 . In general, EMBED Equation.3 where EMBED Equation.3 is the Kronecker delta. We can look at the same result in terms of matrix. The EMBED Equation.3 roots of unity can be used to form an EMBED Equation.3 x EMBED Equation.3 matrix whose EMBED Equation.3 entry is EMBED Equation.3 When EMBED Equation.3 EMBED Equation.3 From above, the columns of this matrix are orthonormal and thus the matrix is unitary. In fact, this matrix is precisely the discrete Fourier transform. III Define EMBED Equation.3 (6) Show EMBED Equation.3 is a norm. In complex inner product spaces, the norm of a complex-valued function f is defined by EMBED Equation.3 To show EMBED Equation.3 is a norm, we must show that: (a) EMBED Equation.3 EMBED Equation.3 (b) EMBED Equation.3 (c) EMBED Equation.3 (d) EMBED Equation.3 Proof of (a): EMBED Equation.3 , by definition EMBED Equation.3 . Proof of (b): EMBED Equation.3 , by definition EMBED Equation.3 EMBED Equation.3 , by definition EMBED Equation.3 , by property of complex number EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 , by property of complex number EMBED Equation.3 and EMBED Equation.3 is constant. EMBED Equation.3 Taking a square root of both sides, EMBED Equation.3 EMBED Equation.3 , by the property of square root, EMBED Equation.3 , iff EMBED Equation.3 . EMBED Equation.3 . Proof of (c): EMBED Equation.3 Prove EMBED Equation.3 . EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 . Prove EMBED Equation.3 . EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 . Proof of (d): EMBED Equation.3 and EMBED Equation.3 are complex valued functions and can be viewed just as a regular complex number EMBED Equation.3 and EMBED Equation.3 . From the properties of inner product on a complex vector space, we know that EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Cauchy-Schwarz Inequality for complex inner product spaces EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 , by (1) EMBED Equation.3 , by (2) EMBED Equation.3 , by (3) Since EMBED Equation.3 is a conjugate of EMBED Equation.3 , when these complex number are summed, imaginary part of EMBED Equation.3 and EMBED Equation.3 would be equal to 0 and the outcome is composed of twice the real part of the complex number. So we can rewrite EMBED Equation.3 as EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 By (4), EMBED Equation.3 EMBED Equation.3 Therefore, EMBED Equation.3 , by (4) EMBED Equation.3 Take a square root of both sides, we obtain, EMBED Equation.3 IV Now we define Fourier transform of the functions in H. Definition: Fourier transform of f is defined EMBED Equation.3 where EMBED Equation.3 and EMBED Equation.3 vary from 0 to p-1 because the domain of function in H is EMBED Equation.3 . Show EMBED Equation.3 Before proving above statement, we would like to do some preliminary computation using p = 3, we compute EMBED Equation.3 and EMBED Equation.3 assuming EMBED Equation.3 is true. EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 By the properties of EMBED Equation.3 we can rewrite the above expression as EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Now let EMBED Equation.3 EMBED Equation.3 Let us look at the exponents of EMBED Equation.3 whose terms involve EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 This means that the terms 1, EMBED Equation.3 , and EMBED Equation.3 are distinct roots. Therefore, by theorem, the sum of roots of unity is zero. The same can be said with the terms that involve EMBED Equation.3 . So the above expression becomes, EMBED Equation.3 EMBED Equation.3 Next, let EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Let us look at the exponents of EMBED Equation.3 whose terms involve EMBED Equation.3 . EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Again, each term in the equation turned out to be distinct. The same can be said with the terms with EMBED Equation.3 . So the equation with EMBED Equation.3 becomes, EMBED Equation.3 EMBED Equation.3 . Lastly, we let EMBED Equation.3 . EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Let us look at the exponents of EMBED Equation.3 whose terms involve EMBED Equation.3 . EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 So each term that involves EMBED Equation.3 is distinct; therefore, the sum of roots of unity is zero: that is, EMBED Equation.3 . Expressing the term EMBED Equation.3 in terms of trigonometric function, we get EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Since the same can be said with the terms with EMBED Equation.3 , our equation becomes, EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Now we are ready to work on the general case. Fourier transform of EMBED Equation.3 is defined EMBED Equation.3 By the properties of EMBED Equation.3 we can rewrite the expression as, EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Notice that if EMBED Equation.3 , terms that involve EMBED Equation.3 become zero as we have seen in example. For example, each term that involves EMBED Equation.3 is distinct because operation involved here is modular arithmetic: therefore, the sum of the root of unity with order p becomes zero. The same thing can be said for the terms with EMBED Equation.3 . Thus, EMBED Equation.3 . When EMBED Equation.3 , the terms that involve EMBED Equation.3 become zero for the same reason, leaving only EMBED Equation.3 terms of EMBED Equation.3 . Therefore, EMBED Equation.3 . Thus, EMBED Equation.3 . (8) Show EMBED Equation.3 Let EMBED Equation.3 . Then, EMBED Equation.3 So EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Notice that EMBED Equation.3 . So EMBED Equation.3 EMBED Equation.3 (9) Show Parseval identity, i.e., EMBED Equation.3 . EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 Let EMBED Equation.3 Then, EMBED Equation.3 if EMBED Equation.3 EMBED Equation.3 if EMBED Equation.3 Hence, EMBED Equation.3 . (10) Show a few numerical examples. EMBED Equation.3 Here we would like to present how the concept of Fourier Transform is used in signal processing through a technique called denoising. The problem in signal processing is that during the transportation of the signal through the devices or air, a signal picks up some noise. The result is well known to all of us who have had problems with cell phone signals. Denoising is a procedure of eliminating the noise from signals. In our examples, we analyze the known signal to which random noise was added. Applying Fourier transform to the noisy data, we reconstruct the original example by first keeping only the highest frequencies and discarding other frequencies by setting them to 0. This procedure is what we call denoising. Example 1: In this example, we took signals which are a superposition of two waves with frequencies at 3 and 37. P is 512. Figure 1 shows the noisy data. Figure 1 Figure 2 below shows the Fourier transform of the data (FFT). Observe high peaks at the frequencies 3 and 37 and symmetrically at 475 and 509, which indicate the main parts of the signal to be cosine waves with frequencies 3 and 37. All the other Fourier transform values are small, and they are coming from the noise. We set them to 0, and then we reconstruct the signal by using the inverse of FFT. Figure 2 Figure 3 below shows the reconstructed signal (in blue) together with original data. Notice that the reconstructing signal avoids high peaks and outliners. Figure 3 Figure 4 shows the reconstructing signal alone. Notice regularity in signal and patterns. Figure 4 The next two figures, Figure 5 and Figure 6, show the reconstruction of the signal without denoising (in red) and the comparison of two signals respectively. Figure 5 Figure 6 We can easily see irregularities in the noisy reconstruction and the lack of uniform patterns. The denoising procedure is quite an effective method in eliminating noise from signals when the noise is not substantial, i.e., when in FFT figure, one can easily read the frequencies of the original signal. But sometimes that is not possible. When that is the case, different methods have to be used. Example 2: In this example, we took a signal which consists of cosine wave with frequency 45. P is 512. Figure 7 shows the noisy data. Figure 7 Figure 8 below shows the Fourier transform of the data. Observe high peaks at the frequencies 45 and symmetrically at 465, which indicate the main part of the signal to be cosine waves with frequencies 45. All the other Fourier transform values are small, and they are coming from the noise. We set them to 0 and reconstruct the signal using the inverse of Fourier Transform. Figure 8 Figure 9 below shows the reconstructed signal together with original data. Notice that the reconstructing signal avoids high peaks and outliners. Figure 9 Figure 10 shows the reconstructing signal alone. Notice regularity in signal and patterns. Figure 10 The next two figures, Figure 11 and Figure 12, show the reconstruction of the signal without denoising and the comparison of two signals respectively. Figure 11 Figure 12 Summary In I, we reviewed what the roots of unity is and the properties of EMBED Equation.3 before plunging into the main problems. As we walked through the problems, we realized that these two notions are essential parts of Fourier Transform and its inverse. When we investigated the properties of EMBED Equation.3 , which is a set of roots of unity, we discovered that EMBED Equation.3 is not only a group but also isomorphic to EMBED Equation.3 . In II, we found that EMBED Equation.3 is an inner product in the complex vector space where EMBED Equation.3 , which is a set of all functions on EMBED Equation.3 into EMBED Equation.3 . When we viewed the roots of unity as a member of H, we discovered that they are orthogonal and in terms of matrices, it formed an orthonormal matrix which is precisely the discrete Fourier transform. At the end of II, we learned that EMBED Equation.3 is a norm. In III, we defined the Fourier transform and its inverse and showed some important properties: EMBED Equation.3 EMBED Equation.3 and EMBED Equation.3 . These are the consequence of the fact that EMBED Equation.3 .are orthogonal. Implication to high school mathematics education Main notions that are strongly related to high school mathematics education are: Modular arithmetic When we learn division in school, we tend to put too much emphasis on the quotient itself or the procedure itself. Through this experience, we realize the importance of encouraging students to interpret division in many different ways. For example, it would be more eye opening and interesting if we viewed division as another way of subtraction just as multiplication can be seen as another way of addition. Moreover, it is even more fun to introduce the notion of modular arithmetic by giving students every day problems such as “What month will it be 25 months from now if it is now December? Modular arithmetic is an abstraction of a method of counting that we often use. It may be a little difficult for students to systematically understand this concept; however, it would benefit them greatly when they pursue higher mathematics. What EMBED Equation.3 means For students to understand Euler’s formula, we need to encourage students to view trigonometry not only as the tool to find out the measures of sides of polygons as we often do, but also to relate it to cords, angles, and points on the unit circle. complex numbers We ought to rigorously study what EMBED Equation.3 means. Also, to deepen the understanding of complex numbers, we might want to assign students similar reading projects we have conducted in our class. Bibliography Anton, Howard. Elementary Linear Algebra. 7th ed. New York: John Wiley & Sons, Inc., 1994 Edwards, Charles Henry. Penney, David E. Differential Equations & Linear Algebra. New Jersey: Prentice-Hall Inc., 2001. Gallian, Joseph A. Contemporary Abstract Algebra. 5th ed. New York: Houghton Mifflin Company, 1996. Herstein, I.N. Abstract Algebra. 3rd ed. New Jersey: Prentice-Hall, Inc., 1996. Maor, Eli. Trigonometric Delights. New Jersey: Princeton University Press, 1998. Nahin, Paul J. An Imaginary Tale. New Jersey: Princeton University Press, 1998. Prestini, Elena. The Evolution of Applied Harmonic Analysis. New York: Birkhauser, 1996. Walker, James S. Fast Fourier Transforms. 2nd ed. New York: CRC press, 1996 HYPERLINK "http://en.wikipedia.org/wiki/Parseval%27s_identity" http://en.wikipedia.org HYPERLINK "http://mathworld.wolfram.com/" http://mathworld.wolfram.com/ HYPERLINK "http://www.everthing2.com" http://www.everthing2.com "#$'45\]bmn P S ¦ § ¾ Ä × ä ò # % & 5 6 C K Q öíáØÍ¿¶ªžªžö•Œˆ„ˆ€ˆ„ˆ|ˆxˆtˆpˆlˆlˆlˆhˆdˆhGhH;šhòµh¾ Jh9jyh¸Ïh[a¶h,…hA}ºhdr1hG5CJaJhòÌ5CJaJhGhOv_5CJ aJ hGh85CJ aJ hòÌ5CJ aJ hòÌhG5CJ(H*aJ(hòÌ5CJ(H*aJ(hòÌ5CJ(aJ(hGh85CJ(aJ(h85CJ$aJ$hOv_5CJaJ' $%&'5?J\]^_`abc÷÷À!¥÷À!¥÷À!¥÷À!¥÷À!¥ïÀ!ÔïÀ!ÔïÀ!ÔïÀ!ÔïÀ!ÔïÀ!Ô÷À!ët÷À!¼÷À!¼÷À!¼÷À!¼÷À!¼÷À!¼÷À!¼÷À!¼÷À!¥÷À!¥÷À!¥÷À!¥÷À!¥êÀ!¥gdòÌ$a$gd8$a$gdÌ1ÎàXýXýýcdefghijklm§ Å Æ KLefúÀ!¥úÀ!¥úÀ!¥úÀ!¥úÀ!¥úÀ!¥úÀ!¥úÀ!¥úÀ!¥úÀ!¥úÀ!¬ò À!¬æÀ!¬ÒÀ!v:ÆÀ!v:¶ÿÀ!:É ªÀ!v:$ Æða$gdA}º$ Æð„Ð^„Ða$gdA}º$ Æð@a$gdA}º$ Æð@„Ð„Ð]„Ð^„Ða$gdÑhØ$„Ð^„Ða$gdA}º$a$gdA}ºgdòÌQ b i } … ‘ © ² ç ?@B|ˆ»ÁÃ"#%&4<Sx}¤¥´¶·¿ - 6 > T p t v w y € ‚ ¶ º Ä Å Æ Ñ Ú üøüôðìèìäðàüÜàØÔØÐØôØôØàÌàÌÜÌäøÜøäøÌÈÀ¸È´È´ü´°¨´°¨´è´ÌÐ’hÑhØhtHìCJaJhÑhØhç9eCJaJhU,hU,H*hÂ 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