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________________________________________________ | SPIA | | An Interactive Guide To | | Signal Processing | | | | USER'S GUIDE | | | | Moonshadow Software | | 5016 Castlewood Drive | | San Jose, CA 95129 | | (408)-446-2459 | | | | (C) Moonshadow Software 1986,1987 | ------------------------------------------------ License Information Non-registered users are granted a limited license to use SPIA on a trial basis for the purpose of determining whether SPIA is suitable for their needs. Use of SPIA, except for this limited purpose, requires registration. Use of non-registered copies of SPIA by any person, business, corporation, government agency or other entity is strictly forbidden. Registration permits a user the license to use SPIA on a single machine. All users are granted a limited license to copy SPIA only for the trial use of others subject to the above limitations, and also the following: * SPIA must be copied in unmodified form, complete with the full documentation (including this license information). * No fee, charge or other compensation may be accepted or requested by any licensee. * SPIA may not be distributed in conjunction with any other product. Operators of electronic bulletin board systems may post SPIA for downloading by their users only as long as the above conditions are met. WARRANTY Moonshadow Software makes no representations or warranties with respect to the contents hereof and specifically disclaims any implied warranties of merchantability or fitness for any particular purpose. Further, Moonshadow Software reserves the right to revise this publication and software and to make changes from time to time in the content hereof without obligation of Moonshadow Software to notify any person of such revision or changes. SPIA is a trademark of Moonshadow Software. ORDERING INFORMATION A SPIA registration licenses you to the use of the product on a regular basis. Registration includes mailed notification of updates and update privileges for a nominal fee. To register send * $79.95 which includes the typeset manual, telephone support, the latest version on disk, and update privileges. Please add $5 shipping, $10 outside U.S. CA residents add sales tax. In addition, evaluation disks are available for $10. These disks do not include registration. Please use the enclosed order form when placing an order. ORDERS OUTSIDE THE US: Please send checks drawn on US banks in US dollars. In addition, make sure $10 shipping is included for any items ordered. ORDER FORM Send to: Moonshadow Software 5016 Castlewood Drive San Jose, CA 95129 (408)-446-2459 Please send me the latest releases of: ____ SPIA Evaluation Disk....................... @ $ 10.00 ea $ ______ (includes program and manual on disk, no registration) ____ SPIA Registration Package.................. @ $ 79.95 ea $ ______ (includes registration, program, typeset manual, telephone support, update privileges) Subtotal ______ (CA residents add sales tax) Tax ______ Total $ ______ Name: ____________________________________________________________ Company: ____________________________________________________________ Address: ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ Phone: _________________________ Comments: ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ Where you heard about/ received your copy: _____________________________________________________ INTRODUCTION SPIA is a mathematical scratchpad for functions. SPIA provides an interactive environment in which to study functions (like sin, cos, etc.) and signal processing techniques (like Fourier transform, convolution, etc.). SPIA is easy to use (really!), and serves a variety of needs. From the high school student learning for the first time about graphing functions and trigonometry to the practicing engineer interested in the power spectrum of a function. SPIA is actually an interpretive language where instead of the scalar variables found in BASIC and FORTRAN one has "function" variables. As a result, one enters algebraic statements very similar to what would be used to manipulate equivalent functions on paper. The user is able to construct functions from that of a predefined set and manipulate them and examine the result graphically. SPIA uses algebraic logic. The following pages are an abbreviated form of the typeset manual available upon registration. SPIA is designed to be intuitive in nature, and we invite you to "jump in". SPIA works the way you might with paper and pencil. SPIA is a electronic scratchpad for mathematical functions and operations. INSTALLATION REQUIREMENTS SPIA requires an IBM PC/XT/AT/System/2 or compatible and a graphics adapter. SPIA also requires 256 K bytes of RAM. The graphics standards supported are: * Hercules monochrome * IBM Color Graphics Adapter (CGA) * IBM Enhanced Graphics Adapter (EGA) * IBM Video Graphics Array Adapter (VGA) Although SPIA does not require a math coprocessor (8087/80287/80387), if your machine has one it will use it. Because SPIA is very math intensive, a math coprocessor will improve its performance significantly. On an IBM PC/XT with no math coprocessor the time for a Fourier transform is about 20 sec. SPIA can be run from a floppy or hard disk. BACKUP COPIES Before configuring or installing SPIA on your computer, make a copy of the SPIA SHAREWARE distribution disk. We encourage you to make copies of this diskette to share with your colleagues. Once SPIA is configured for your computer the SPIA disk is customized and cannot be used on another computer without running the included RECONFG program. So to prevent confusion, please distribute the unconfigured original SPIA SHAREWARE distribution diskette. CONFIGURATION and INSTALLATION To configure and install SPIA on your computer simply insert the SPIA SHAREWARE distribution disk in drive A and type "CONFIG" (without the quotes). For example: A>CONFIG [Enter] The floppy disk is customized for your computer and if requested a customized copy is also transferred to the hard drive in C:\SPIA. RUNNING SPIA Once SPIA has been configured it is run by typing RUNSPIA. For a floppy based system simply insert the configured diskette in drive A (or B) and type "A:[Enter]" and then "RUNSPIA". On a hard drive one needs to first change to the SPIA subdirectory and then type RUNSPIA. For example: C>CD \SPIA [Enter] C>RUNSPIA [Enter] USING SPIA SPIA is essentially an interpretive programming language, in which the variables are functions (vectors) and the operators are the basic mathematical operations with a few special ones thrown in. By taking the supplied functions and operators (and a little ingenuity) one can custom build an enormous array of functions upon which to experiment. The commands which invoke these functions and operations are entered in the command window at the bottom of the screen. The commands are pretty much what one might write on paper to algebraically set up the mathematical operation. The commands and functions are detailed in the command summary section. SPIA's working screen is divided into 3 functional windows: The display window, the message window, and the command line window. ________________________________________________ | | | | | | | DISPLAY WINDOW | | | | | | | ------------------------------------------------ | MESSAGE WINDOW | ------------------------------------------------ | COMMAND WINDOW | ------------------------------------------------ THE DISPLAY WINDOW. It is here that help screens are displayed and the graphics representation of the functions are displayed. When displaying functions, there are 3 work areas used: A, B, and C. This allows the viewing of 3 functions at a time. Each function has as its range -2 to +2, with 0 marked on the x axis. The ordinate range is marked on the y axis each time a function is displayed. Unless specified all functions and their resulting manipulations are placed in the A work area for display. In a programming sense it is assigned to A. For example: SINX + COSX gets the predefined functions sin(x) and cos(x), adds them together and places the result in the A work area. This is equivalent to saying A = SINX + COSX USING SPIA - continued- Once a work area has been assigned a result, it can be used in subsequent expressions, for example ABS(A) takes the absolute value of A, which currently has the result of SINX+COSX. Notice too, that this expression places the results of ABS(A) in A! (since there is no explicit assignment) So, one can recursively assign work areas, much like that of variables in BASIC and FORTRAN. To place the results in the other work areas simply make the assignments, like so A = SINX B = SINX + COSX C = ABS(B) The above command sequence would allow us to see the progression of our operations graphically, instead of writing over our intermediate steps. One other aspect of the display work areas bears mentioning - the display of imaginary functions. Since SPIA can operate with complex functions (ie. functions with both real and imaginary parts) the display area is structured so that one can display either the real or imaginary parts independently. This is accomplished by typing either IA, IB, and IC in the command window to view the imaginary parts of the complex functions A, B or C. IA, IB and IC are also used to view the phase parts of A, B and C if the functions have been converted to their amplitude/phase form by the AMP() function. More about complex functions later. THE MESSAGE WINDOW. As its name implies, this window provides you with messages pertaining to the program's operation. There are 4 types of messages: Error messages Informational messages Prompts Mode messages THE COMMAND LINE WINDOW. This is a three line scrolling window which displays your commands. GENERAL NOTES SPIA has a dictionary of functions (nouns) and operators (verbs). With these it is possible to construct new functions upon which further operations can be carried out. The command lines are entered and parsed one line at a time based on a set of precedence rules. The available functions, operators, and commands are described in detail on the following pages and a quick summary can be invoked during program operation by typing HELP. SPIA is meant to be exploratory in nature, so don't hesitate just trying things out. The functions in SPIA are defined over the range of x = -2 to x = +2, eg. SINX, the sin(pi*x), ranges over -2pi to +2pi. Each function contains 256 points. PARSING The parser in SPIA will parse (that is grammatically resolve the command line into a collection of functions and operations on those functions) most reasonable algebraic command lines. If there is any doubt as to the order of operations, the liberal use of parentheses will force the order to your liking, eg: B = A + B * C will be done as A + (B*C) B = (A + B) * C will be done as (A+B) * C The precedence of operators is: 'unary' > 'binary' = '*' = '/' > '+' = '-' where unary refers to operators with only one argument, like FT(A), IFT(A), SQRT(A), ABS(A), and binary refers to operators like CONV and CORR that require two (A CONV B). When operator precedence is equal, the left most operator has precedence. If you are unsure of how the computer interpreted your command - watch the message window. This will reflect the order of operation. For example A+B*C will flash in succession "evaluating product", "evaluating sum". Like english (or at least like most programming languages) SPIA has a grammar, and while it is reasonably easy to trip it up, it is also reasonably easy to deliver commands that do what you had wanted! Somewhere in between an adventure game and a calculator. Currently, the unary minus is not supported. It is easy to work around. For example: (0 - 1) * GAUSS GENERAL NOTES - continued - COMPLEX ARITHMETIC Because Fourier Transforms generate complex functions, there is the ability to manipulate and display these complex functions in SPIA. There are two modes within SPIA: COMPLEX ON and COMPLEX OFF. These two commands can be typed to force the mode of arithmetic calculation. Basically if complex arithmetic is on, then addition, subtraction, multiplication and division take into account the imaginary parts of the functions. Complex arithmetic is automatically turned on when taking the FT of some function, and it is automatically turned off when the inverse FT is taken. Again, these modes can be manually invoked. It is important when doing arithmetic in the frequency domain that the complex mode be on, otherwise the imaginary component of the result of the operation always has a zero component. COMMAND SUMMARY MATHEMATICAL OPERATIONS ----------------------- ABS(A)......the absolute value of the function in A. AMP(A)......converts real/imaginary to amplitude phase of A. CONJ(A).....the complex conjugate of A. A CONV B....performs the convolution integral of A with B. A CORR B....performs the cross correlation integral of A with B, DERIV(A)....the difference between the ith and (i+1)th points for all points, with the final point being equal to the differential before it. Hence, errors can accumulate in the last point. EXP(A)......the exponential of each point in A. FORCE(A)....forces the endpoints of A = 0, thereby eliminating end effects in the Fourier transform. FT(A).......the Fourier Transform of A. The imaginary part can be displayed by IA. Complex arithmetic mode automatically turned on. IFT(A)......the inverse Fourier Transform of A. Complex mode turned off. INTEG(A)....the integral of A. LOG(A)......the log base 10 log of each point in A. LN(A).......the log base e of each point in A. MIRROR(A)...the right side of a function mirrored into the eft. MSHIFT(A)...shifts the function A left (minus) by 10%. RE(A).......converts function from amplitude/phase space to real/imaginary. SHIFT(A)....shifts the function A right (plus) by 10%. SMOOTH(A)...3 point smooth of A. SQRT(A).....the square root of each point in A. TEN(A)......the power of A to the base 10, for each point in A. A + B.......sums each point of A and B respectively. A - B.......subtracts each point of B from each point of A. A * B.......multiplies each point of A and B respectively. A / B.......divides each point of A by each point of B, respectively. A ** n......takes each point of A and raises it to the n-th power. Note, number can be any real number, but it cannot be another function. Ie. SINX**COSX is not allowed. A = B.......takes each point in B and assigns it to A. COMMAND SUMMARY - continued - DEFINED FUNCTIONS ----------------- n...........any number can be input as a function, this has the effect of assigning each point of that function to be that number. eg. B=17.5 assigns each point of B to be 17.5, (SINX+1)/NOISE*10 adds 1 (one) to each point of the function SINX, and divides it by the function NOISE times 10. A,B,C.......these are user defined functions, having zero values until assigned. B and C must be explicitly assigned, as in B=COSX, but A is implicitly assigned by typing COSX. IA,IB,IC....these are the imaginary counterparts to A,B, and C. COSX........cosine of pi*x where x:-2 to +2 DEL0........delta function at x=0 EXPMX.......the exponential of the absolute value of x:-2 to +2 exp(-abs(x)). GAUSS.......the gaussian. defined as exp(-pi*(x**2)) HEAVY.......the Heavyside step function, defined as zero for x<0 and 1 for x>0. III.........sampling function, delta functions at regular intervals. II..........pair of delta functions at x=1,-1. LOREN.......the lorentzian, mean of zero, halfwidth of 1. NOISE.......pseudo random noise of intensity -0.1 to +0.1 (approximately 10% of many of the above functions). NOTCH.......defined as 1 for abs(x) > 0.1, and 0 for abs(x) < 0.1, excellent for applying filers to FT's. RECT........rectangle function, 0 for abs(x) > .5, 1 for abs(x) <= .5 SIGN........defined as -1 for x<0 and +1 for x>=0. SINX........sin(pi*x) over -2pi to +2pi. Also supported are SIN2X thru SIN9X, where SIN9X is sin(9*pi*x) SINCX.......defined as sin(pi*x)/(pi*x), sinc(0) = 1. Also SINC2X thru SINC9X. TRI.........the triangle function, defined as 1-abs(x) for abs(x) < 1 and 0 elsewhere. X...........ramp from -1 to +1. COMMAND SUMMARY - continued - COMMANDS -------- COMPLEX [ON/OFF]..used to control complex arithmetic mode, when complex mode is on the operations +,-,*, and / are done with both the imaginary and real parts of the functions. Note the complex annunciator is turned on in complex mode. EXIT..............exit program. FAST [ON/OFF].....determines if convolution/correlation is done with integer arithmetic (fast) or floating point (slow). Functions must be prepared before use by multiplying by the desired precision. Eg: 100*SINX allows fast convolution to within 3 significant figures. FILE..............reads in user acquired data files. Text files (ASCII) can be read, where the numbers are one per line (with no commas etc.) and can be integer, floating point or exponential, eg: 32 .73 8.1E-05 HELP..............displays a quick summary of commands. PLOT..............plots screens. PRINT.............prints the screen. RESTORE...........restores previously saved functions. SAVE..............saves current set of functions. SUPPRESS [ON/OFF].suppresses digital error propagation. Default is on. QUIT..............same as exit. EXAMPLES FOURIER FILTERING. One of the novel techniques available in Fourier space is frequency filtering. In this case we will filter out the high frequency component from our signal. The theory being that the low frequency information is what the signal of interest is comprised of and the high frequency information is purely noise. The truth of this statement can vary drastically from situation to situation, lets look at two of them. CASE I (1) Create a gaussian with noise, place it in C for reference GAUSS + NOISE (places the result in A) C = A (2) Take the transform of the resultant function, lets also save it for further manipulation FT(A) B = A (3) Apply a notch filter to the transform, create it first in A, then multiply the transform, B, by it (this applies the filter) and finally take the inverse transform. 1 - NOTCH (like a very narrow square function) A * B (note this is complex multiplication) (A now holds the filtered transform) IFT(A) You'll find that the resultant function in A is a smoothed version of GAUSS+NOISE in C. What would have happened if the filter function to be applied was RECT? Also try adding 10*NOISE, instead of just NOISE, to GAUSS, to give a signal to noise ratio of 1:1 and repeat the filter exercise. EXAMPLES - continued - CASE II (1) Let's apply this technique to another function. Create a train of peaks like so: SIN5X**10 (2) Reproducing the steps (1) to (3) in case I, only using this function as the base A + NOISE C = A FT(A) B = A (1 - NOTCH)*B IFT(A) Apparently not an appropriate filter to apply in this case, due to the higher frequency content of our desired signal! Try it with RECT as the filter function. Note the above examples could have been created with far fewer steps by simply applying the operations successively: IFT((1 - NOTCH)*B) This concludes the abbreviated on-disk SPIA manual. Please pass this software on. And if you find SPIA useful, please do register it. Remember, registration pays for the continuing development of this software and software like this. Thank you for your support.