Functions Menu

The user loads two complex arrays f1 and f2 as real/imaginary parts (or as magnitude/phase) from the FUNCTIONS MENU, which is selected by typing F from any menu. The default array to load is the complement of that most recently loaded or viewed, but either may be selected. Until the array is initialized (values set to zero), functions are added to existing data, thus allowing functions to be modified (e.g., truncated) after operations have been performed. All functions generated and displayed in SIGNALS are single-precision floating-point numbers. The largest and smallest allowed magnitudes are approximately 10+38 and 10-45, though the dynamic range is truncated somewhat during calculations to avoid known compatibility problems with some computers and graphics displays. Double-precision computations are used within some operations (such as the transformations, raising amplitudes to a power, and computing statistics). Error handling routines control most over- and underflow conditions, but some computations involving weighted sums of the array (such as in the FFT) may fail if array amplitudes are saturated. However, this likely will never occur in normal use.

Data arrays are loaded in sequence:

At each step, several functions may be summed to "build" the data arrays.

If arrays are loaded as MAGNITUDE-PHASE, remember that all operations and plots assume REAL-IMAGINARY format; option R in the OPERATIONS menu converts MAGNITUDE-PHASE to REAL-IMAGINARY before other processing.

The desired function is selected by typing the highlighted letter, e.g., type S to add SINUSOIDs (sines and/or cosines). The user is prompted to select the parameters of the function; defaults are indicated in brackets [ ]. In this example, the user first selects the number of SINUSOIDs to add (the default is always 1); Note that typing zero or a negative number at this prompt returns to the FUNCTIONS menu -- this allows recovery from mistyping the function selection. Next, the user is prompted for the period of the SINUSOID b [N/2], the phase (in degrees) at the origin [0], and the amplitude [1]. Fractions are allowed ("2/5" is read as 0.4), and reciprocals are indicated by a leading slash / ("/2" is read as 0.5). After the parameters of a function have been selected, the user is returned to the FUNCTIONS menu to add more terms if desired. If the desired function is real valued, the user may type <ESC> to return immediately to the MAIN MENU. Once the REAL/MAGNITUDE array has been filled, the user types a carriage return <CR> (i.e. the "ENTER" key) to produce the same menu of functions; this time a modulating function may be generated which is multiplied by the original REAL PART/MAGNITUDE. If no modulation is desired, type <CR> to continue to the IMAGINARY PART/PHASE, where the same sequence of operations may be performed. You MUST type <CR> from the MODULATION MENU to perform the multiplication; if you <ESCape>" to the MAIN MENU before typing <CR>, the functions will not be multiplied. If a modulation function is started and then the array is re-initialized (zeroed) and not replaced, the initial array will be multiplied by a null array when continuing to the IMAGINARY array. An option ("|") is available in the MODULATION MENU for the REAL PART/MAGNITUDE to compute the absolute value of the array. This is used when data is entered as MAGNITUDE-PHASE.

The options in the FUNCTIONS array are:

Z         Zero Array:       f(n) = 0(n)

S        Sinusoid      Equation for Sinusoidal Function      

Graph of Sinusoidal Function

D        Discrete Dirac Delta Function      Equation for Delta function

Graph of Delta Function

R        Rectangle Function: f(n) = A RECT(( n - n0 ) / b)      Equation for Rectangle Function

Graph of Rectangle Function

T        Triangle Function: f(n) = A TRI(( n - n0 ) / b)     Equation for Triangle Function

Graph of Triangle Function

S        SINC Function:  f(n) = A SINC(( n - n0 ) / b)     Equation for SINC function

Graph of SINC function

G        Gaussian Function: f(n) = A GAUS(n - n0)      Equation for Gaussian Function

Graph of Gaussian Function

G        SIGNUM Function: f(n) = A SGN(n - n0)      Equation for SIGNUM function

Graph of SIGNUM Function

P        STEP Function:    A STEP[n] = 1/2 (1 + SGN[n])  

Graph of STEP Function

J        Bessel Function of order n       Notation for Bessel Function

Bessel Function of Order 0

Bessel Function of Order 1

1      Constant: f(n) = A 1(n)

C      Chirp Function (Quadratic-Phase Sinusoid):       Equation for Chirp Function

Graph of Chirp Function

F      Fresnel Zone Plate:         Equation for Fresnel Zone Plate

Graph of Fresnel Zone Plate

M      COMB Function:        Equation for COMB Function

V      Square Wave:       Equation for Square Wave

Graph of Square Wave

3     3-Bar Chart

K      DC Blocking Filter:    f(n) = 1 - RECT(n/b)

E      Exponential Function:        f(n) = e-n/b STEP(n)

B      BESINC Function       Equation for BESINC Function

L      Line segments

O     Power Series Terms: f(n) = A ( n / b )p 

-     Interpolation Functions:

> Special Functions:

N      Noise from Probability Distributions p(f)
        G        Gaussian Noise of selectable mean and standard deviation.
        U        Uniform Distribution with selectable mean and range
        2         Uniform Distribution over a range of 2 pi
        B        Binomial Distribution of rate p
        N        Boltzmann Distribution (negative exponential)
        P        Poisson Distribution with selectable rate
        S        Poisson Point Process (impulses at Poisson-random intervals)
        L        Lorentzian distribution of selectable rate (also called a Cauchy distribution)
        R        Rayleigh distribution with selectable parameter

        Graphs of Realizations of Noise Distributions

W    Data Windows
        B         Bartlett window (triangle weighting)
        H         Hanning window (cosine weighting)
        M         Hamming window (biased-cosine weighting)
        S         Half-cycle Sine window
        L         Blackman-Harris window (3 cosine factors)
        K         Blackman window (2 cosine factors)
        3         Half-cycle Sine3 window
        G         Gaussian window
        C         Cosine2 window
        W         Welch window (parabolic falloff)
        E         Extended Cosine bell window (unity over 80% of data)
        R         Rectangle window (unity over 96% of data)
        P         Parzen window
        Y         Cauchy window (similar to Lorentzian)

          Equations and Graphs of Window Functions

   OPTIONS in OPERATIONS MENU (suggest that "Image Autoload" be turned off!)
   OPTIONS in PLOT MENU                (suggest that "Image Autoload" be turned off!)