{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "Normal" -1 256 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 281 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 286 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 291 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 296 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 297 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 300 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 301 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 304 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 305 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 306 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 307 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 308 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 310 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 311 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 312 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 313 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 314 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 315 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 316 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 317 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 318 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 319 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 320 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 321 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 322 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 323 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 324 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 325 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 326 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 327 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 328 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 329 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 330 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 331 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 332 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 333 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 334 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 335 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 336 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 337 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 338 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 339 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 340 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 341 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 342 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 343 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 344 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 345 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 346 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 347 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 348 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 349 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 350 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 351 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 352 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 353 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 354 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 355 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 356 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 357 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 358 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 359 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 360 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 361 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 362 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 363 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 364 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 365 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 366 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Fixed Width" 0 17 1 {CSTYLE "" -1 -1 "Co urier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 17 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "" 19 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 17 "" 0 "" {TEXT -1 39 " Paradigms in Physics: Oscil lations" }}{PARA 17 "" 0 "" {TEXT -1 0 "" }}{PARA 17 "" 0 "" {TEXT -1 73 " 2000 Oregon State University version July 2 0, 2000" }}{PARA 17 "" 0 "" {TEXT -1 6 " " }}{PARA 18 "" 0 "" {TEXT 303 17 "Fourier Integrals" }}{PARA 256 "" 0 "" {TEXT -1 39 "by P hilip J. Siemens and Jason Janesky " }}{PARA 19 "" 0 "" {TEXT -1 43 "P hysics Department, Oregon State University" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 " Introduction " }}{PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 258 1 " " }{TEXT 257 571 " The goal of this worksheet is to become famili ar with the relation between a function and its Fourier transform. Ce rtain analytic changes to the original function will produce recogniza ble effects in the Fourier-transformed function. This worksheet is de signed to illustrate these properties. It should help to build an int uition for the complementary aspects of these dual representations. A s an analogy, you can consider the functions to represent the physical time pulse of an electric signal and the Fourier transform as the fre quency spectrum of that signal." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 304 7 "CHOOSE " }{TEXT 308 4 "PLOT" }{TEXT 349 1 " " }{TEXT 309 6 "WINDOW" }{TEXT 350 10 " FROM THE " }{TEXT 310 25 " OPTIONS/PLOT DISPLAY MENU" }{TEXT 311 22 " OF YOUR MAPLE SCREEN." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 63 "WHEN WOR KING THROUGH THE THEORETICAL PARTS OF THIS WORKSHEET, " }}{PARA 0 "" 0 "" {TEXT 312 33 "KEEP SEVERAL GRAPHS UP AT ONCE. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 313 69 "THE IDEA IS TO SEE THE FUNCTION AND ITS FOURIER TRANSFORM TOGETHER. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 91 "ALSO IT IS IMPORTANT TO \+ SEE HOW DIFFERENT ALGEBRAIC CHANGES PRODUCE DIFFERENT EFFECTS. SO:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 306 28 " DO N'T CLOSE THE GRAPHS," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 307 47 " KEEP THEM ALIGNED TO COMPARE THEM VISUALLY" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 21 "PLAN OF THE WORKSHEET" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "The worksheet contains several examples of various functi onal forms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "The best introduction is to start with the exponential pulse." }}{PARA 0 "" 0 "" {TEXT -1 78 "The rectangular, Gaussian and Lorentzia n cases are included as extra examples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "The section on Harmonic Response of \+ an LRC Circuit builds on the Exponential Pulse." }}{PARA 0 "" 0 "" {TEXT -1 113 "It can help you predict quantitatively the current in a \+ series circuit in response to a harmonic input voltage. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 " The Exponential Pulse" }{TEXT 330 4 " \+ " }}{PARA 0 "" 0 "" {TEXT 359 111 "The exponential pulse is a typica l waveform for an electronic circuit. It can be parametrized by an am plitude " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 22 " and a decay cons tant " }{XPPEDIT 18 0 "beta" "6#%%betaG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f(t):=A*exp(-beta*t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "For nu merical examples we need to choose specific values of the parameters. \+ For example," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A:=1; beta:= 1;" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 331 26 "A plot of the function is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "pl ot(f(t),t=-3..3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 " " {TEXT -1 69 "This function is not a pulse! At early times it is big , not small! " }}{PARA 257 "" 0 "" {TEXT -1 88 "Close the window for \+ this plot to discard it. It was a good first try but not a keeper." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 35 "We nee d to start the pulse at time " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT 346 21 "To do this we use the " }{TEXT -1 1 " " }{TEXT 348 58 "step function, built into Maple as th e Heaviside function:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f(t):=A*exp(-beta*t)*Heaviside(t);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 347 18 "A plot o f this is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "plot(f(t),t=-3..3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "That's more like it!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 339 54 "PUT THIS PLOT IN TH E UPPER LEFT CORNER OF YOUR SCREEN." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 258 "" 0 "" {TEXT -1 41 "FOURIER TRANSFORM OF AN EXPONENTIAL PU LSE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 332 114 " The Fourier transform built into Maple is like the Engineer's fourier \+ transform in the notes, except for a factor " }{XPPEDIT 18 0 "sqrt(2*P i)" "6#-%%sqrtG6#*&\"\"#\"\"\"%#PiGF(" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=fou rier(f(t),t,w)/sqrt(2*Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 340 81 "Notice the form of the transformed function. \+ It is called a Lorentzian function." }}{PARA 0 "" 0 "" {TEXT 353 2 " \+ " }}{PARA 0 "" 0 "" {TEXT 352 200 "The transformed function now has re al and imaginary parts. In other words the amplitude corresponding \+ to a specific frequency is no longer descibed by a single real number \+ but requires two entries," }{TEXT 364 5 " i.e." }{TEXT 365 19 " a comp lex number. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 351 12 "The plot is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{Re(G(w)),Im(G(w))\},w=-10..10);" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Which li ne is the real part, and which is the imaginary part?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "Notice that one of the se is an odd function of w, and the other is an even function. Why?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 333 55 "PUT T HIS PLOT ALONGSIDE THE PREVIOUS PLOT TO THE RIGHT." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 354 101 "Another way to represen t this complex number is by the amplitude and phase of each Fourier co mponent." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "AmpG(w):=sqrt(Re(G(w))^2+Im(G(w))^2);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "PhaG(w):=arctan(Im(G(w))/Re(G(w)));" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(\{Am pG(w),PhaG(w)\},w=-10..10);" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Which line is the amplitude, and which is the p hase?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "PUT THIS PLOT TO THE RIGHT OF THE PREVIOUS PLOT,, " }}{PARA 0 "" 0 " " {TEXT -1 133 "so that you have a row of windows along the top of the screen with the function and two views of its fourier transform, all \+ in a row." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 355 67 "TAKE A MINUTE AND RESIZE THE PREVIOUS WINDOWS TO MAKE THEM ALL FIT." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 26 "WIDTHS AND COMPLEMENTARITY" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 334 77 "Now let's see what effect widening the p ulse has on the transformed function." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "We input a smaller value of beta:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "beta:=0.5;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f(t):=A*exp(-beta *t)*Heaviside(t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "plot(f(t),t=-3..3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 341 56 "PUT THIS ONE BELOW THE F IRST PLOT TO COMPARE THE WIDTHS." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=fourier(f(t),t,w)/sqrt (2*Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 335 69 "Notice the change in the transformed function as compared with abo ve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{Re(G(w)),Im(G(w))\},w=-10..10);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "AmpG(w):=sqrt(Re(G(w))^2+Im(G(w))^2);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "PhaG(w):=arctan(Im(G(w))/Re(G(w))); " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(\{AmpG(w),PhaG(w)\},w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 59 "RESIZE THESE PLOTS AND LINE THEM UP B ELOW THE PREVIOUS SET." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 336 262 "Notice:that making the pulse wider in one variable makes it narrower as a function of the other variable. This means th at a longer pulse can be constructed using a narrower band of frequenc ies. We say that the widths of the pulse and its fourier transform ar e " }{TEXT 356 13 "complementary" }{TEXT 357 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 29 "OSCILLATIONS AND TRANSLATIONS " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "What \+ happens if the exponential also oscillates?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Let's go back to the original w idth:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "beta:=1;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 74 "First, let's try multiplying by an exponential wit h angular frequency w1 :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f(t):=A*exp(-beta*t)*Heaviside(t)*e xp(I*w1*t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 157 "We need to choose a value for w1. To choose the oscil lation period comparable to the decay time we have to choose a fairly \+ large value of w1 since it takes " }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"#\" \"\"%#PiGF%" }{TEXT -1 43 " radians to make an oscillation. Let's try " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " w1:=5;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(\{Re(f(t)),Im(f(t)) \},t=-3..3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 342 41 "PUT THIS PLOT IN THE LOWER LEFT CORNER. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=f ourier(f(t),t,w)/sqrt(2*Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 337 69 "Notice the change in the transformed fun ction as compared with above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{Re(G(w)),Im(G(w))\},w =-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "G_amplitude(w):=sqrt((Re(G(w)))^2 + (Im(G(w)))^2); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "G_phase(w):=arctan(Im(G(w))/Re( G(w)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(\{G_amplitude(w),G_ phase(w)\},w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "RESIZE THESE GRAPHS AND LINE THEM UP AT THE BOTTOM O F THE SCREEN." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "What has happened to the Fourier transforms? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "They have been \+ translated by an amount w1: " }}{PARA 262 "" 0 "" {TEXT -1 0 "" }} {PARA 263 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "GNew(w) = GOld(w-w1) " "6#/-%%GNewG6#%\"wG-%%GOldG6#,&F'\"\"\"%#w1G!\"\"" }}{PARA 0 "" 0 " " {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "This is easy to under stand analytically, since" }}{PARA 261 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "GNew(w)=int(FOld/sqrt(2*Pi),t=0..infinity)" "6#/-%%GNew G6#%\"wG-%$intG6$*&%%FOldG\"\"\"-%%sqrtG6#*&\"\"#F-%#PiGF-!\"\"/%\"tG; \"\"!%)infinityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(I*w1*t)*exp(-I*w *t)" "6#*&-%$expG6#*(%\"IG\"\"\"%#w1GF)%\"tGF)F)-F%6#,$*(F(F)%\"wGF)F+ F)!\"\"F)" }{TEXT -1 1 "=" }{XPPEDIT 18 0 "Int(FOld/sqrt(2*Pi),t=0..in finity)" "6#-%$IntG6$*&%%FOldG\"\"\"-%%sqrtG6#*&\"\"#F(%#PiGF(!\"\"/% \"tG;\"\"!%)infinityG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-I*(w-w1)*t )" "6#-%$expG6#,$*(%\"IG\"\"\",&%\"wGF)%#w1G!\"\"F)%\"tGF)F-" }}{PARA 261 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 82 "To accomplis h this we needed an oscillation of the phase of FNew compared to FOld. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "What \+ happens if we just make the function oscillate without making it compl ex?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Fo r example, we can multiply by " }{XPPEDIT 18 0 "sin(w1*t) = (exp(I*w1* t)-exp(-I*w1*t))/2/I" "6#/-%$sinG6#*&%#w1G\"\"\"%\"tGF)*(,&-%$expG6#*( %\"IGF)F(F)F*F)F)-F.6#,$*(F1F)F(F)F*F)!\"\"F6F)\"\"#F6F1F6" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 117 "By expressing the sine in term s of exponentials, we can see that we will get two terms, translated l eft and right by " }{XPPEDIT 18 0 "w1" "6#%#w1G" }{TEXT -1 28 " . Let 's see how this works" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f(t):=A*e xp(-beta*t)*Heaviside(t)*sin(w1*t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(t),t=-3..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " G(w):=fourier(f(t),t,w)/sqrt(2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{Re(G(w)),Im(G(w))\},w=-10..10);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "As expected we see twin patterns at " }{XPPEDIT 18 0 "w1" "6#%#w1G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "-w1" "6#,$%#w1G!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 71 "Note that the roles of real and imaginary parts have be en interchanged." }}{PARA 0 "" 0 "" {TEXT -1 29 "This is due to the fa ctor of " }{XPPEDIT 18 0 "I" "6#%\"IG" }{TEXT -1 80 " in the denominat or of the expression for sine in terms of complex exponentials." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "How does \+ this look in terms of amplitude and phase?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "G_amplitude(w):=sq rt((Re(G(w)))^2 + (Im(G(w)))^2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "G_phase(w):=arctan(Im(G(w))/Re(G(w)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(\{G_amplitude(w),G_phase(w)\},w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The amplitude is eas y enough to understand, but what about the phase?" }}{PARA 0 "" 0 "" {TEXT -1 37 "It is discontinuous, jumping between " }{XPPEDIT 18 0 "Pi /2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "-P i/2" "6#,$*&%#PiG\"\"\"\"\"#!\"\"F(" }{TEXT -1 4 " . " }}{PARA 0 "" 0 "" {TEXT -1 64 "The discontinuity happens when the real part goes th rough zero. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 138 "The denominator in the arctan changes sign, so the tange nt of the angle changes from a large positive value to a large negativ e value (or " }{TEXT 358 10 "vice versa" }{TEXT -1 67 "). The arctan \+ function interprets this as a change of phase from " }{XPPEDIT 18 0 " Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "- Pi/2" "6#,$*&%#PiG\"\"\"\"\"#!\"\"F(" }{TEXT -1 23 " . Angles larger \+ than " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 15 " but less than " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 41 " have th e same tangent as angles between " }{XPPEDIT 18 0 "-Pi/2" "6#,$*&%#PiG \"\"\"\"\"#!\"\"F(" }{TEXT -1 65 " and 0. So we can also think of the phase as increasing beyond " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\"\"\" \"\"#!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "To see how this works, consider the following function:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "test(t):=arctan(tan(t ));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 343 157 "We know that the operations of tan followed by arctan should negate e ach other, and the plot should just be a straight line. Let's see how Maple handles it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "plot(test(t),t=-5..5);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 344 236 "So from this we can sur mise that the discontinuity we saw above is just an image of a continu ously increasing function. We have to train our eyes to piece the sec tions back together. This is easier than training the computer to do \+ it!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 338 153 " The preceding analysis referred to electrical pulses as a physical ana logy. In fact the result is very general, and applies to many physica l systems. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 345 483 "An especially interesting example appears in quantum me chanics. The pulse function can be seen as a wavefunction in position space and the Fourier transform would be the wavefunction in momentum space. The inverse relation between their widths as known as the Hei senberg Uncertainty Principle. These connections are not magically co njured, but come out of the math that is used in analyzing these situa tions. The physical interpretation of the math has occupied many grea t minds." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 " Harmonic Response of an LRC Circuit" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "with(inttrans): unprotect(Beta):" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "WHEN YO U ARE USING THIS SECTION IN CONNECTION WITH LABORATORY MEASUREMENTS, \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "CHOOS E PLOT DISPLAY/INLINE FROM THE OPTIONS MENU. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "THIS WILL MAKE IT CONVENI ENT TO SAVE A RECORD OF YOUR WORK." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 74 "The response of an electric circuit to an impulse input is described by a " }{TEXT 360 17 "response function" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 211 "For example, if the input is a voltage pulse applied to \+ a pair of terminals, and the measured quantity is the current through \+ the terminals, the result is given in terms of the current-voltage res ponse function " }{XPPEDIT 18 0 "ChiIV(t)" "6#-%&ChiIVG6#%\"tG" } {TEXT -1 50 ". This means that, for an input pulse of voltage " } {XPPEDIT 18 0 "V0" "6#%#V0G" }{TEXT -1 27 " lasting a very brief time \+ " }{XPPEDIT 18 0 "dt" "6#%#dtG" }{TEXT -1 9 " at time " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\"!" }{TEXT -1 30 ", the current at a later time " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 3 " is" }}{PARA 265 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 2 " ." }}{PARA 266 "" 0 "" {XPPEDIT 18 0 "I(t) = V0*dt*ChiIV(t)" "6#/-%\"IG6#%\"tG*(%#V0G\"\"\"%# dtGF*-%&ChiIVG6#F'F*" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "For the series LRC \+ circuit, the simplest response function is the charge-voltage response function, " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ChiqV(t):=AqV *sin(Omega*t)*exp(-Beta*t)*Heaviside(t);" }}}{PARA 268 "" 0 "" {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 " The current-voltage response function is its derivative, " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "ChiIV(t):=diff(ChiqV(t),t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Insert values for the constants \+ which apply to a circuit you are interested in." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "You can take them from o bservations, or calculate them from the values of the resistance, capa citance and inductance. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 71 "Which source of constants have you chosen? _______ _____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "BE SURE TO USE SUITABLE UNITS, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 " For example, if th e time is in milliseconds, then the frequency is in radians per millis econd, which is the same as kiloradians per second. What are the unit s of Aqv?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "Omega:=1;" "6#>%&OmegaG\"\"\" " }{TEXT -1 1 ";" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Beta:=0.2;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "AqV:=1;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Check the plot of the response function:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(ChiIV(t),t=-1..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "The ste ady-state response of an electric circuit excited by a harmonic input \+ signal is described by the " }{TEXT 361 21 "generalized impedance" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Z" "6#%\"ZG" }{TEXT -1 18 " of the circ uit. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "For example, if the input signal is an applied voltage " }{XPPEDIT 18 0 "V(t)=cos(omega*t+phi)" "6#/-%\"VG6#%\"tG-%$cosG6#,&*&%&omegaG\" \"\"F'F.F.%$phiGF." }{TEXT -1 35 " then the current will be given by " }}{PARA 269 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I(t)=Re(V0*exp(I* phi)*ZIV(omega)*exp(I*omega*t)" "6#/-%\"IG6#%\"tG-%#ReG6#**%#V0G\"\"\" -%$expG6#*&F%F-%$phiGF-F--%$ZIVG6#%&omegaGF--F/6#*(F%F-F6F-F'F-F-" } {TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "ZIV(omega)" "6#-%$ZIVG6#%&omegaG" }{TEXT -1 50 " is the current-voltage generalized impedance, or " } {TEXT 362 10 "admittance" }{TEXT -1 68 ", which is related to the resp onse function by the Fourier transform" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 270 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "ZIV(t)" "6#-%$ZI VG6#%\"tG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(ChiIV(t)*exp(-I*omega *t),t=-infinity..infinity)" "6#-%$IntG6$*&-%&ChiIVG6#%\"tG\"\"\"-%$exp G6#,$*(%\"IGF+%&omegaGF+F*F+!\"\"F+/F*;,$%)infinityGF3F7" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "We let Maple evaluate the integral: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ZIV(omega):=simplify(eval(fourier(ChiIV(t),t,omega))) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Let's plot the function. First c hoose a range of omega which is appropriate to your choice of units.: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "omegamax:=3;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot(\{Re(ZIV(omega)),Im(ZIV(omega))\},omega=0..omega max);" }}}{PARA 0 "" 0 "" {TEXT -1 61 "Which line is the real part, an d which is the imaginary part?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Let's also look at the amplitude a nd phase:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "AmpZIV(omega):=sqrt(Re (ZIV(omega))^2+Im(ZIV(omega))^2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "PhaZIV(omega):=arctan(Im(ZIV(omega))/Re(ZIV(omega))):" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot(\{AmpZI V(omega),PhaZIV(omega)\},omega=0..omegamax);" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Which line is the amplitu de, and which is the phase?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "6#%#%?G" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "These predictions star ted from the response function observed as the circuit's response to a n impulse input. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "We can also formulate similar predictions from circuit t heory, based on the nominal values of the circuit elements." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "For a series ci rcuit, the " }{TEXT 363 9 "impedance" }{TEXT -1 58 " ZVI is the sum of the impedances of the circuit elements:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ZVI(omega):=R + I*ome ga*L + 1/(I*omega*C);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The admittance is the reciprocal of the impedance:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ZIV(omega):=1/ZVI(omega);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 42 "Let's make a plot of this prediction, too :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Firs t input the values of the circuit elements:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "R:= 0.4;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "L:=1;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "C:=1;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot(\{Re(ZIV(omega)) ,Im(ZIV(omega))\},omega=0..omegamax);" }{XPPEDIT 19 1 "" "6#%#%?G" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Which lin e is the real part, and which is the imaginary part?" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Let's also look \+ at the amplitude and phase:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "AmpZ IV(omega):=sqrt(Re(ZIV(omega))^2+Im(ZIV(omega))^2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "PhaZIV(omega):=arctan(Im(ZIV(omega))/Re(ZIV(omega) ));" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot(\{AmpZIV(omega),PhaZIV(omega)\},omega=0..omegamax);" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Which line is \+ the amplitude, and which is the phase?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "6#%#%?G" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "How does this pre diction compare with the prediction from the Fourier transform of the \+ response function?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "PRINT THE MAPLE SHEET NOW TO RECORD YOUR WORK." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "You should use it in y our lab report, and " }{TEXT 366 40 "plot your observations on the sam e axes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "When you do, be sure to use the same units for theory and observat ion!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 " The Rectangular Pulse" }}{PARA 0 "" 0 "" {TEXT 259 446 " The rectangula pulse is just a function defined to have a constant value over a defined range, while outside of this range it i s zero. This can be represented by the piecewise operation in Maple. \+ Instead we will use the Heaviside theta-function defined in Maple, wh ich corresponds to standard mathematical notation. to become more fami liar its features. The pulse we will use is defined below to be 1 whe n its argument is between -1 and 1:" }}{PARA 0 "" 0 "" {TEXT 260 1 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f(t):=1*Heaviside(1+t)*Heavi side(1-t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 18 "A plot of this is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(t),t=-10..10);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 292 54 "PUT THIS PLOT IN \+ THE UPPER LEFT CORNER OF YOUR SCREEN." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 25 "The \+ Fourier transform is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=fourier(f(t),t,w)/sqrt(2*Pi);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 92 "Notice t he form of the transformed function. It is called a Fresnel function. The plot is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(G(w),w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 55 "PUT THIS PLOT ALONGSIDE THE PR EVIOUS PLOT TO THE RIGHT." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 264 73 "Let's see what effect widening the pulse has on t he transformed function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f(t):=1*Heaviside(5+t)*Heaviside(5- t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(t),t=-10..10) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 294 56 "PU T THIS ONE BELOW THE FIRST PLOT TO COMPARE THE WIDTHS." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=f ourier(f(t),t,w)/sqrt(2*Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 265 69 "Notice the change in the transformed fun ction as compared with above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(G(w),w=-10..10);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 162 "Notice: the wider the pulse in one space the narrower it is in the other. Th is means that a longer pulse can be constructed using a narrower band \+ of frequencies." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 42 "Let's try a displacement along the t axis." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f(t): =1*Heaviside(-3+t)*Heaviside(5-t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(t),t=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 295 41 "PUT THIS PLOT IN THE LOWER LEFT COR NER. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 315 67 "TAKE A MINUTE AND RESIZE THE PREVIOUS WINDOWS TO MAKE THEM ALL FIT ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=fourier(f(t),t,w)/sqrt(2*Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 69 "Notice the change in \+ the transformed function as compared with above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{Re(G( w)),Im(G(w))\},w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 326 "The transformed function now has real and ima ginary parts. In other words the amplitude corresponding to a speci fic frequency is no longer descibed by a single real number but requir es two entries, i.e. a complex number. Another way to represent this \+ complex number is by the amplitude and phase of each Fourier component ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "G_amplitude(w):=sqrt((Re(G(w)))^2 + (Im(G(w)))^2);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "G_phase(w):=arctan(Im(G(w) )/Re(G(w)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(\{G_am plitude(w),G_phase(w)\},w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 472 "The sawtooth pattern may look a bit s trange. This is not related to the input function. Instead, it is a \+ result of how Maple deals with the function arctan. It is an unavoid able result of the fact that the computer has to choose one of the man y angles with the same tangent. The same problem arises in defining t he functions arcsin and arccos. Maple chooses what is called the pri ncipal value. For the arctan, the resulting angle will always be betw een the values " }{XPPEDIT 18 0 "-Pi" "6#,$%#PiG!\"\"" }{TEXT 317 5 " \+ and " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT 318 4 "; " }{XPPEDIT 18 0 "-pi" "6#,$%#piG!\"\"" }{TEXT 319 11 " < arctan(" }{XPPEDIT 18 0 "t heta" "6#%&thetaG" }{TEXT 320 5 ") < " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT 321 49 " . Your pocket calculator has a similar feature. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 316 131 "To unde rstand the result of this prescription, we will take a small sidestep. Let's look at how Maple deals with something we know:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "test(t) :=arctan(tan(t));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 157 "We know that the operations of tan followed by arctan s hould negate each other, and the plot should just be a straight line. \+ Let's see how Maple handles it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(test(t),t=-5..5);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 302 231 "So from this we can surmise that the sawtooth we saw above is just an image o f a continuously increasing function. We have to train our eyes to pi ece the sections back together. This is easier than training the comp uter to do it!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 " The Gaussian Pulse" }}{PARA 0 "" 0 "" {TEXT 271 82 " We will proceed in the same fashion in our study of the Gaus sian function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "f(t):=1*exp(-(t/1)^2);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "plot(f(t),t=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 25 "The Fourier transform is :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=fourier(f(t),t,w)/sqrt(2*Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 88 "Notice that we get ba ck a Gaussian in the transformed space, but with a different norm." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(G(w),w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 59 "AGAIN USE THE SAME ARRANGEMENT OF PLOTS TO SEE THE EFFECTS." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 80 "Let's broaden the Gaussian in the t-space. Can you gues s the effect in w-space?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f(t):=1*exp(-(t/5)^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(t),t=-10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=fourier(f(t),t,w)/sqrt(2*Pi); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 33 "How is this different from above?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(G(w),w=-10..10);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 73 "Notice h ow in one space the pulse broadens and in the other it narrows. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "INTERPRETING THESE OBSERVATIONS" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 322 328 "The inv erse relationship between widths of functions in Fourier-adjoint repre sentations was discussed in depth in connection with the discovery of \+ Quantum Mechanics in the 1920's. The connection was formulated mathem atically by Werner Heisenberg and interpreted phyisically by Niels Boh r and Erwin Schroedinger, among others. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 323 170 "Soon afterwards, during the 1930 's, the implications of this relationship for electrical circuits and \+ communications were explored by Claude Shannon and John von Neumann." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 59 "Next let 's try, as before, a displacement along the t-axis." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f(t):=1*ex p(-((t-5)/1)^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f( t),t=-10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=fou rier(f(t),t,w)/sqrt(2*Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 90 "Notice the effect on the transformed result. \+ How does it change the transformed function?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{Re(G(w)),Im (G(w))\},w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 297 64 "Again this can be reperesented by a complex amplitude and phase." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "G_amplitude(w):=sqrt((Re(G(w)))^2 + (Im(G(w)))^2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "gG_phase(w):=arctan(Im( G(w))/Re(G(w)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(\{ G_amplitude(w),G_phase(w)\},w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 169 "To show the duality of these spa ces, let's multiply the original function by a complex exponential. W hat effect do you think this will have on the transformed function?" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f(t):=1*exp(-(t/1)^2)*exp(I*3*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{Re(f(t)),Im(f(t))\},t=-10..10);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 39 "Looking at this a s amplitude and phase:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f_amplitude(t):=sqrt((Re(f(t)))^2 + (Im(f(t)))^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f_phase( t):=arctan(Im(f(t))/Re(f(t)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(\{f_amplitude(t),f_phase(t)\},t=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 299 32 "As for the transfor med function:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=fourier(f(t),t,w)/sqrt(2*Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 72 "Notice where the 3 enters into the transformed function. And the graph:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot( G(w),w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 55 "We could try both effects at once and see the outcome: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f(t):=1*exp(-((t-5)/5)^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(t),t=-10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=fourier(f(t),t,w)/sqrt(2*Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 84 "Compare this next gra ph with the ones above to distinguish the simultaneous effects." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{Re(G(w)),Im(G(w))\},w=-10..10);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "G_amplitude(w):=sqrt((Re(G(w)))^2 + (Im(g(w)))^2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "G_phase(w):=arctan(Im(G (w))/Re(g(w)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(\{G _amplitude(w),G_phase(w)\},w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 " The Lorentzian Pulse" }} {PARA 0 "" 0 "" {TEXT 283 235 " This analysis will be very similiar to the preceding as far as the algebraic transformations. Therefor: \+ more actions and less words. The function is proportional to the real part of the Fourier transform of the exponential pulse:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f(t): =1/(t^2+1^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(t), t=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT 284 93 "This looks close to but is not quite like a Gaussian. It has l onger tails. The transform is:" }{TEXT 285 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=fourier(f(t),t,w)/sqrt(2*Pi);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Interpretation of this result" }}{PARA 0 "" 0 "" {TEXT 324 62 "The two terms can be combined into one compact representation:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "G(w)=sqrt(Pi/2) *e^(-abs(w))" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 325 89 "Converse ly, the Heaviside (piecewise) function is a way to avoid absolute-valu e notation." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(G(w),w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 38 "Instead of broadening the func tion of " }{XPPEDIT 328 0 "t " "6#%\"tG" }{TEXT 327 79 " we will make \+ it narrower to get a better look at the decaying exponentials in " } {XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT 326 7 "-space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f(t):=1/(t^2 +(.2)^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(t),t=-1 0..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=fourier(f( t),t,w)/sqrt(2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot (G(w),w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 287 38 "Notice again the broadening effect in " }{XPPEDIT 18 0 " w" "6#%\"wG" }{TEXT 329 7 "-space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 288 25 "Now for the displacement:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f(t):=1 /((t-5)^2+1^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(t ),t=-10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=four ier(f(t),t,w)/sqrt(2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{Re(G(w)),Im(G(w))\},w=-10..10);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "G_amplitude(w):=sqrt((Re(g(w)))^2 + (Im(G(w)))^2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "G_phase(w):=arctan(Im(G (w))/Re(G(w)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(\{G _amplitude(w),G_phase(w)\},w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 289 62 "Let's broaden the peak here to see \+ the effect. Can you guess?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f(t):=1/((t-5)^2+3^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(t),t=-10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=fourier(f(t),t,w)/sqrt(2*Pi); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{Re(G(w)),Im(G(w) )\},w=-10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "G_amplit ude(w):=sqrt((Re(G(w)))^2 + (Im(G(w)))^2);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "G_phase(w):=arctan(Im(G(w))/Re(G(w)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(\{G_amplitude(w),G_phase(w)\}, w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 41 "What if we shifted the peak further over?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f(t):=1/((t- 8)^2+3^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(f(t),t=- 10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "g(w):=fourier(f (t),t,w)/sqrt(2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plo t(\{Re(G(w)),Im(G(w))\},w=-10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "G_amplitude(w):=sqrt((Re(G(w)))^2 + (Im(G(w)))^2);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "G_phase(w):=arctan(Im(G(w) )/Re(G(w)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(\{G_am plitude(w),G_phase(w)\},w=-10..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 300 177 "Let's see whether multiplying the Lor entzian by a complex exponential has the effect of shifting in the tra nsformed space, as we saw with the Gaussian and the Rectangular Pulse. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f(t):=exp(-I*8*t)/(t^2+1^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f_amplitude(t):=sqrt((Re(f(t)))^2 + (Im(f(t)))^2);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f_phase(t):=arctan(Im(f(t) )/Re(f(t)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot(\{f_am plitude(t),f_phase(t)\},t=-10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G(w):=fourier(f(t),t,w)/sqrt(2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(G(w),w=-10..10);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 291 71 "Using cut and paste yo u can explore more relationships at your leisure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "15" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }