1 What You Need To Know About Basic Maths

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1 1 Wht You Need To Know About Bsic Mths Before we cn strt on ny of the more interesting stuff, I thought I d mke sure tht everyone hs the bsic mthemticl bckground necessry to red nd understnd everything in the rest of the book. One of the gols of the book is to minimise the mount of mthemtics required, but there s no getting wy from it: solid grounding in mths is essentil to understnding the principles nd techniques in modern communictions engineering. This isn t mths textbook I m not going to try nd tech nyone mths, but I thought it would be useful to summrise the mths you ll need to know for the rest of the book, so you cn red over these first few chpters, nd then feel confident tht you re well equipped for the journey into the wonderful world of communictions engineering. Or t lest, you ll know wht you re missing, nd need to ctch up on. This first chpter of the wht you should know section contins brief summry of the bsic essentil mths knowledge most people do in school, nd gives me chnce to introduce some of nottion I ll be using the rest of the book. Redy? Then let s begin. 1.1 Vribles, Functions nd Signls A vrible is quntity tht we cn t ssign fixed vlue to. So we cll it something like x or y, nd then try to write equtions tht let us work out the vlue of x or y. Most often, I ll write vribles in itlics 1. For exmple, if I hd vrible x, nd I knew tht x + 3 = 9, then I could work out tht in this cse x = 6. Sdly, most equtions ren t tht esy. Functions re opertions tht trnsform one or more numbers or vribles into other numbers or vribles. Most of the functions we ll be meeting hve single output, but some of them tke more thn one input (note tht n input to function is often clled n rgument). For exmple, consider function tht tkes one rgument, nd outputs double the input. I could write this s: f ( x) x (0.1) where f(x) is this function. When the vrible x = 3, we just substitute ll instnces of x with the number 3, nd we cn clculte: f (3) 3 6 (0.) Another exmple: function tht tkes two rguments, nd outputs the squre of difference between the two rguments: so tht if x = nd y = 3, we get: g( x, y) x y (0.3) g(,3) (0.4) 1 The exceptions re vector or mtrix vribles which I ll write in bold. More bout vectors nd mtrices lter. 006 Dve Perce Pge 1 8/01/009

2 Notice tht this prticulr function hs the property tht g(x, y) = g(y, x). Functions with this property re sid to be commuttive. Most functions don t hve this property. A signl is function of time only tht outputs single vlue. For exmple, the voltge on wire or the sound pressure in the ir crrying sound. They re usully written like s(t) to show they re functions of time Symmetry nd Periodicity Some functions hve prticulr properties of interest. For exmple, function is sid to be even-symmetric if for ll vlues of x, f(x) = f( x), nd odd-symmetric if for ll vlue of x, f(x) = f( x). Even symmetric f(x) = f( x) Odd symmetric f(x) = f( x) Figure 1-1 Even nd Odd Functions Notice tht ll odd-symmetric functions must be zero t the origin, in other words f(0) = 0. Interestingly, ny function of single vrible cn be expressed s the sum of n even-symmetric function nd n odd-symmetric function. It s done like this: let o(x) be n odd-symmetric function, nd e(x) be n even symmetric function. Then, consider: ( ) ( ) ( ) ( ) o( x) f x f x nd e( x) f x f x It s strightforwrd to prove tht o(x) is n odd-function, since: f ( x) f ( x) f ( x) f ( x) o( x) o( x) (0.5) nd even esier to prove tht e(x) is n even-symmetric function (I ll leve tht one to you). It s lso esy to show tht: f ( x) f ( x) f ( x) f ( x) o( x) e( x) f ( x) (0.6) Although the single vlue might be vector with severl components. There s more bout vectors in lter chpters, don t worry if tht doesn t men nything to you yet. 006 Dve Perce Pge 8/01/009

3 This turns out to be very useful result. Another interesting property of some functions is periodicity. A periodic function is one tht obeys f(t) = f(t + T) for ll t, where T is known s the period of the function 3. T Periodic f(t) = f(t + T) Note tht if f(t) = f(t + T), then this lso implies tht f(t) = f(t + T), since: f ( t T ) f t T T f ( t T ) f ( t) (0.7) nd the sme is true for f(t) = f(t + 3T) nd in fct tht f(t) = f(t + nt) for ny integer vlue of n, however the period is tken to be the smllest possible positive offset for which the function repets. 1. Logrithms The logrithm is prticulrly useful function of single input, which provides single output vlue. The only importnt point is tht the input must be positive number, since it s difficult to tke the logrithm of negtive number 4. The logrithm function is usully written s log b (x), where b is the bse of the logrithms. The logrithm of number x is the power to which given number (known s the bse) must be rised to give x. It s usully written s log b (x), where b is the bse of the logrithms. Therefore, b log ( x ) b x (0.8) For exmple, the logrithm (bse 10) of 1000 is 3, or in mthemticl nottion log 10 (1000) = 3, 3 becuse Another exmple, the logrithm (bse ) of 64 is 6, or in mthemticl nottion log (64) = 6, 6 becuse 64. Logrithms re useful, since they llow the time-consuming process of multipliction to be replced by the much simpler process of ddition. This is chieved by noting the most useful property of logrithms: the sum of the logrithms of two numbers is the logrithm of the product of the two numbers. This is esy to prove: 3 Most periodic functions of interest re functions of time, so I ve used the nottion f(t) here insted of f(x). 4 It s not impossible to tke the logrithm of negtive number, but the result is complex number, not rel number. More bout this in the chpter on complex numbers. 006 Dve Perce Pge 3 8/01/009

4 log( x) log( y) log( x) log( y) log( xy) bse bse bse xy bse (0.9) A similr result is true for the subtrction of two logrithms: this provides simple wy to do the even more time-consuming process of dividing two numbers: log( x) log( y) log( x) log( y) log( x) bse x log( x / y) log( y) bse bse bse bse (0.10) bse y Of course, you hve to find the logrithm of the numbers first, nd then tke the power of the bse to the logrithm of the nswer to get the finl nswer. Logrithms hve some more useful nd importnt properties s well, for exmple if you know the logrithm bse A of number x, nd you need the logrithm bse B, then: log B ( x) log A( x) (0.11) log ( A) nd if you know the logrithm bse A of number x, nd you need the logrithm of x, then: becuse: B log ( x ) log ( x) (0.1) A log ( x ) log ( ) x log ( x) A A A A A x x A A (0.13) 1.3 Algebr Algebr is n essentil tool in engineering in prticulr the bility to simplify expressions, nd to express formuls in terms of whichever vribles re of most interest. It s importnt to be very fmilir with results such s: nd to recognise results such s: x y x y x y (0.14) x 3 y 3 x y x xy y which cn be proved by multiplying out the right-hnd-side: (0.15) x y x xy y x 3 x y xy x y xy y x x y xy x y xy y (0.16) x 3 3 y 006 Dve Perce Pge 4 8/01/009

5 Chnging the vrible in n expression is n essentil skill for exmple you ll need to be ble to chnge the formul: Q M ( n 1) D Bn (0.17) into M Q( D 1) n Q( B 1) (0.18) It s done by doing the sme opertion to both sides of the eqution, until the only term left on the left-hnd side is the one you wnt, in this cse n. Here, this is ccomplished by multiplying both sides by (n 1) + D + Bn, then subtrcting Q(D 1), nd finlly by dividing by Q(B+1) Qudrtic Equtions A prticulrly importnt result in lgebr is the formul for qudrtic equtions 5 : x bx c 0 b b 4c x (0.19) In exmple problems, it s useful to be ble to spot the fctors of simple qudrtic equtions: for exmple, if given n eqution of the form: with prctice it s possible to solve this by inspection, by noting tht: x x 6 0 (0.0) 5 This is derived by process known s completing the squre, nd goes like this: x bx c 0 b b x bx c b b x c 4 b b x c 4 b b 4c x b b 4c x 006 Dve Perce Pge 5 8/01/009

6 x x 6 ( x 3)( x ) (0.1) nd hence tht this is solved by x = 3 or. However, in the rel world, the solutions re rrely tht simple. 1.4 Problems 1) If 3 x 7x 6x 0, wht re the possible vlues of x? ) Given the formul: 3n 4, n 1 solve for ll possible vlues of n. Wht re the lrgest nd smllest possible vlues of n tht stisfy this eqution? 3) Wht is log x (x )? 4) Wht is the result of dividing x 3 y 3 by x y? 5) Two numbers when multiplied together give 31.5, but when dded together give 15. Wht re the two numbers? 6) If log x (15) = 3, wht is x? 7) If log (x) = 4, nd log (y) is, wht is log (xy)? x 8) If 1.05, tke the logrithm of both sides of this eqution, nd hence solve for x. 006 Dve Perce Pge 6 8/01/009