Digital signal processing - dsp

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Dsp00095-preface to digital signal processing - dsp

0.1 Table of contents

0.3 Legacy versus openstax presentation format

0.4 Miscellaneous

Dsp00100-periodic motion and sinusoids

0.1 Table of contents

0.3 Viewing tip

0.6 Discussion and sample code

0.7 Periodic motion

0.8 Motion of a pendulum

0.9 A bag of sand

0.10 The sand traces out a pattern

0.11 A sinusoidal function

0.12 Harmonic motion

0.14 Separate the cosine and sine components

0.15 Plotting separate sine and cosine functions

0.16 Time varying functions

0.17 Sampled data

0.18 The period of the sine and cosine functions

0.19 Different periods or frequencies

0.20 Modified arguments

0.21 Why does pi appear in the arguments ?

0.22 Radians versus cycles

0.23 The horizontal scale

0.24 The argument to the sine and cosine functions

0.25 Where does sin(arg) equal zero ?

0.26 Where are the peaks in the cosine function ?

0.27 Composition and decomposition

0.28 An approximate square waveform

0.29 Successive approximations

0.30 Each curve contains more sinusoidal terms

0.31 Reasonably good approximation

0.32 Each term improves the approximation

0.33 The first five sinusoidal terms

0.34 A side-by-side comparison

0.35 An approximate triangular waveform

0.36 Five sinusoids

0.37 Intermediate waveforms

0.38 Good result with only five terms

0.39 The individual terms

0.41 A practical example

0.42 The frequency filters

0.43 The spectrum analyzer

0.45 Miscellaneous

1 Dsp00104-sampled time series

1.1 Table of contents

1.3 Viewing tip

1.7 Sinusoids, time series, composition, and decomposition

1.8 The notion of sampling analog signals

1.9 What is meant by sampling ?

1.10 Uniform sampling is most common

1.11 Some problems arise

1.12 Reconstruction of the analog signal

1.13 Reconstruction is not always required

1.14 The Nyquist folding frequency

1.15 A brief description

1.16 Some specific numbers

1.17 A few comments about sampling

1.18 What do we really have ?

1.19 Sampled sinusoids

1.20 The number of samples per cycle is important

1.21 The values between the samples

1.22 Estimating the values in between the samples

1.23 A more common representation

1.24 The most common representation

1.25 What happens when the sampling frequency is reduced ?

1.26 Some numbers

1.27 Comparison of frequencies with center frequency

1.28 Reduce the sampling frequency

1.29 Re-plot the sampled data

1.30 Center plot no longer resembles a sinusoid

1.31 The top two sinusoids

1.32 Top sinusoids match bottom sinusoids

1.33 Bottom three plots are correct

1.34 A different approach

1.35 Sampling frequency was four samples per second

1.36 The five sinusoids

1.37 The heights of the spectral peaks

1.38 The locations of the spectral peaks

1.39 Introduce a sampling problem

1.40 Sampling frequency was two samples per second

1.41 The heights of the spectral peaks

1.42 The locations of the spectral peaks

1.43 A problem with the location of two spectral peaks

1.44 The folding frequency

1.45 The peak in the fifth plot

1.46 Back to the case with no problems

1.47 Sampled at four samples per second

1.48 Figure 9 Shows twice the frequency range

1.49 Folding frequency at the center

1.50 Why is it called the folding frequency?

1.51 Peaks below folding frequency are valid

1.52 Normally don't compute the mirror image

1.53 Back to the case with the sampling problem

1.54 Folding frequency is at the center

1.55 Peaks to left correspond to Figure 8

1.56 A mirror image

1.57 Raw data frequency exceeds folding frequency

1.58 Artifacts to the left of the folding frequency

1.59 Normally can't identify artifacts

1.60 Illustrates the folding frequency

1.61 The bottom line

1.62 Using an analog low-pass pre-filter

1.63 Digital re-sampling

1.65 Miscellaneous

2 Dsp00108-averaging time series

2.1 Table of contents

2.3 Some of what you have previously learned

2.4 Viewing tip

2.8 Which screw to turn ...

2.9 The mechanic and the screw

2.10 A very important module

2.11 Turning the screws in DSP

2.12 Computing the average value of a time series

2.13 Decomposition of time series

2.14 A sum of products of sinusoids

2.15 The product of two sinusoids

2.16 The frequencies of the new sinusoids

2.17 Sum and difference frequencies

2.18 An important special case

2.19 Many sinusoidal products

2.20 Product of sinusoids with same frequency

2.21 The ideal average value

2.22 Product of sinusoids with different frequencies

2.23 Ideal average value is zero

2.25 Examples of products of sinusoids

2.26 Multiplying sinusoids with same frequency

2.27 A double-frequency sinusoid

2.28 Half the amplitude with a positive bias

2.29 The sum of two sinusoids

2.30 Product of sinusoids with different frequencies

2.31 The sum of two sinusoids

2.32 Another view of the same data

2.33 More than one cycle

2.34 Averaging can be problematic in this case

2.35 More examples of the products of sinusoids

2.36 A greater frequency difference

2.37 An even greater frequency difference

2.38 What is the average value of a sinusoid ?

2.39 Is that a true statement?

2.40 Sample average values

2.41 The computed average value

2.42 Result converges on the ideal

2.43 An expanded view

2.44 Ideal value is never reached

2.45 A more serious case

2.46 Red curve is the average

2.47 The ideal average value is zero

2.48 Some additional examples of average values

2.49 A short recap before continuing

2.50 The error in the computed average

2.51 Does this matter?

2.52 Spectral analysis

2.53 Several steps are involved

2.54 Spectral data for same frequency but different lengths

2.55 Sum of two sinusoids

2.56 Sinusoids with different frequency differences

2.57 The Fourier transform

2.58 Keeping it simple

2.59 A brief description of the fourier transform

2.60 Why does this work?

2.61 Multiply by the cosine and average the product

2.62 Multiply by the sine and average the product

2.63 What about the other sinusoidal components?

2.64 The average value for non-matching components

2.65 Measurement error

2.66 Let's see some data

2.67 Five sinusoids, same frequency, different lengths

2.68 Extend to 400 samples for computation

2.69 The actual averaging window

2.70 A horizontally-expanded plot

2.71 A spectral line

2.72 Spectra of shorter sinusoids

2.73 Two spectral lines

2.74 Five time series with two sinusoids each

2.75 Spectral analysis using fourier transform

2.76 Discuss the longest time series first

2.77 Resolution for shorter averaging windows

2.78 No peak separation for 80-sample average

2.79 A horizontally-expanded view of the spectra

2.80 Zero-valued points in the spectra

2.81 400 spectral values were computed

2.82 Sampling frequency was one sample per second

2.83 The computational frequency interval

2.84 Let's compute the average of some products

2.85 The average of the product time series

2.86 Product of two cosine functions

2.87 The average of the product time series

2.88 Where do the zero values match?

2.89 Zero values in the spectrum for a sinusoid

2.90 The frequency resolution of the Fourier transform

2.91 Illustration of frequency resolution

2.92 A match for the frequency resolution

2.93 Insufficient frequency resolution

2.94 Sufficient resolution in all five cases

2.95 Adequate frequency resolution in all cases

2.96 Peaks were resolved in all five cases

2.98 Miscellaneous

Java1468-plotting engineering and scientific data using java

0.1 Table of contents

0.3 Viewing tip

0.7 The Graph01Demo program

0.8 Using the plotting program with your data

0.9 The Graph01 program

0.10 Discussion and sample code

0.11 Testing the plotting program

0.12 Using the plotting program

0.13 Plotting your data using Graph01

0.14 Modifying plotting parameters

0.15 The plotting parameters

0.16 The labels on the axes

0.17 Keep the pixels in mind

0.18 The interface named GraphIntfc01

0.19 Implementing graphintfc01

0.20 The method named getnmbr

0.21 The methods named f1, f2, f3, f4, and f5

0.22 One plotting area per method

0.23 How does it work?

0.24 Doesn't know and doesn't care

0.25 Computed "on the fly"

0.26 Returned from an array

0.27 From a disk file, a database, the internet, etc.

0.28 It is up to you

0.29 The class named Graph01Demo

0.30 Defining data-generator classes

0.31 The number of functions to plot

0.32 The method named f1

0.33 The method named f2

0.34 The method named f3

0.35 The method named f4

0.36 The method named f5

0.37 The end of the class definition

0.38 The class named Dsp002

0.39 A DSP example

0.40 White random noise

0.41 A convolution filter

0.42 The convolution filter output

0.43 Time-domain vs. frequency-domain

0.44 Frequency spectrum analysis

0.45 The white noise spectrum

0.46 The filtered noise spectrum

0.47 Two spectral peaks are visible

0.48 What does the symmetry mean?

0.49 An ambiguity in the spectrum analysis

0.50 Let's see some code

0.51 Different from the previous example class

0.52 Basic operation of the program

0.53 Beginning of the class named Dsp002

0.54 Create data arrays

0.55 Beginning of the constructor

0.56 The random noise generator seed

0.57 Create the convolution operator

0.58 Apply the convolution operator

0.59 Compute spectrum of each of two traces

0.60 All results have been computed and saved

0.61 The getNmbr method

0.62 The method named f1

0.63 Convert incoming parameter to an index

0.64 Stay within array bounds

0.65 Methods f2 through f5

0.66 The class named Convolve01

0.67 The discrete Fourier transform (dft)

0.68 Two plotting programs

0.69 The program named Graph01

0.70 Some general comments

0.71 Separate plotting areas

0.72 A noarg constructor is required

0.73 Some methods may not be called

0.74 The visual appearance

0.75 A test class

0.76 Usage instructions

0.77 Plotting parameters

0.78 A new object

0.79 Requires java sdk 1.4 or later

0.80 Will discuss in fragments

0.81 The class named Graph01

0.82 Beginning of the class named GUI

0.83 Beginning of the constructor for the GUI class

0.84 Array to hold Canvas objects

0.85 Routine GUI construction code

0.86 The Canvas objects

0.87 More routine construction code

0.88 Force a repaint

0.89 End of the constructor

0.90 Re-plotting the data

0.91 A new object of the target class

0.92 The remainder of the event handler

0.93 Beginning of the class named MyCanvas

0.94 Floating data vs. pixels

0.95 A simple constructor

0.96 Beginning of the overridden paint method

0.97 Get old coordinate values

0.98 The initial y-coordinate value

0.99 Plot the points

0.100 The drawAxes method

0.101 Drawing tic marks

0.102 The getTheX and getTheY methods

0.103 The test class named junk

0.104 The program named Graph02

0.105 Run the program

0.107 Complete program listings

0.108 Miscellaneous

1 Java1489-plotting 3d surfaces using java

1.1 Table of contents

1.3 Viewing tip

1.6 General Discussion

1.7 Displaying 3D data can be fairly difficult

1.8 Different approaches are available

1.9 A 3D plotting program

1.10 Grayscale plot

1.11 A calibration scale

1.12 Color Shift plot

1.13 More information is conveyed

1.14 The lowest elevations are obvious in the color shift plot

1.15 Four minor peaks are obvious in the color shift plot

1.16 The highest elevation is obvious in the color shift plot

1.17 Quantitative estimates are possible

1.18 Color Contour plot

1.19 Similar to a contour map

1.20 More minor peaks

1.21 The design of this program

1.22 Good quantitative elevation information is available

1.23 The elevations of the minor peaks

1.24 The lowest elevations

1.25 The elevation of the valleys

1.26 A logarithmic conversion

1.27 Flattens the plot

1.28 More sample output

1.29 Six different formats

1.30 Isolates the location of the minima

1.31 The log of the surface is flatter

1.32 Multiple plotting formats are useful

1.33 Extremely easy to use

1.34 Multiple plots

1.36 Constructor

1.37 Parameters

1.42 Display = 0

1.43 Display = 1

1.44 Display = 2

1.45 Display = 3, 4, and 5

1.46 A calibration scale

1.47 Normalization

1.48 For display value of 0 or 3

1.49 For display value of 1 or 4

1.50 For a display value of 2 or 5

1.51 The main method

1.52 A 3D parabola

1.53 The calibration scale

1.54 The images are stacked

1.55 With and without axes

1.56 A window listener

1.58 The program named ImgMod29

1.59 The main method

1.60 Local variables

1.61 The array for the surface elevation data

1.62 The blocksize parameter

1.63 A 3D parabolic surface

1.64 Display six surface images

1.65 Just instantiate an object

1.66 The parameters

1.67 Instantiate six separate objects

1.68 The ImgMod29 class

1.69 Establish display format for log conversion

1.70 The three basic types

1.71 Copy the input elevation data

1.72 Convert to log data if required

1.73 New to j2se 5.0

1.74 Normalize the surface elevation data

1.75 The method named scaleTheSurfaceData

1.76 Create an appropriate pair of Canvas objects

1.77 The interesting code

1.78 Add the Canvas objects to the Frame

1.79 Call the pack method and set the title

1.80 Register an anonymous WindowListener object

1.81 Six different inner classes

1.82 Subclasses of the canvas class

1.83 The getCenter method

1.84 Even or odd is very important

1.85 Grayscale plot format

1.86 The constructor

1.87 Overridden paint method for CanvasType0surface class

1.88 Purpose of overridden paint method

1.89 Instantiate a Color object

1.90 Set colors and draw squares

1.91 Paint a colored square

1.92 Draw the optional red axes

1.93 Beginning of the class named CanvasType0scale

1.94 How it works

1.95 The overridden paint method

1.96 Color Shift plot format

1.97 Beginning of the class named CanvasType1surface

1.98 Overridden paint Method

1.99 Set white and black for max and min values

1.100 Elevations other than the extreme ends

1.101 Processing the lowest range

1.102 Scale the elevation values

1.103 The color aqua

1.104 Processing the other three ranges

1.105 Gradient from aqua to green

1.106 Gradient from green to yellow

1.107 Gradient from yellow to red

1.108 A homework assignment

1.109 The rest of the overridden paint method

1.110 The class named CanvasType1scale

1.111 Color Contour plot format

1.112 The color palette

1.113 The getcolorpalette method

1.114 Rearrange and add new colors

1.115 The CanvasType2surface class

1.116 Quantize the elevation levels

1.117 Set the color value

1.118 The class named CanvasType2scale

1.119 Run the program

1.121 Complete program listing

1.122 Miscellaneous

2 Java1492-plotting large quantities of data using java

2.1 Table of contents

2.3 Viewing tip

2.7 Sample output for PlotALot01 class

2.8 Usage information

2.9 Plotting format

2.10 On to the next page

2.11 Nearly unlimited plotting capacity

2.12 Millions of data values plotted

2.13 Two sample pages

2.14 Two overloaded constructors

2.15 Can change default plotting parameters

2.16 Control over the plotting parameters

2.17 Default plotting parameters in Figure 1

2.18 Explanation of terms

2.19 Location of the stack of plots

2.20 Sample output for PlotALot02 class

2.21 Superimposed data

2.22 Same data, different sign

2.23 Plotting parameters were controlled

2.24 Larger ovals

2.25 Horizontal scaling was greater

2.26 Sample output for PlotALot03 class

2.27 Same data, two colors

2.28 The plotting parameters

2.29 Sample output for PlotALot04 class

2.30 Same data, three colors

2.31 The plotting parameters

2.32 Sample programs

2.33 The class named PlotALot01

2.34 Purpose of the class

2.35 Usage information

2.36 Different plotting objects

2.37 Can plot a large number of data values

2.38 Multiple page objects

2.39 A stack of page objects

2.40 Multiple traces on each page object

2.41 Two overloaded constructors

2.42 Constructor parameters

2.43 Two plotting objects for test purposes

2.44 The data to be plotted

2.45 Transition from trace to trace on the same page

2.46 Transition from page to page

2.47 Proper locations for awt frame under winxp

2.48 Information about plotting parameters

2.49 Overloaded plotdata method

2.50 A windowlistener for program termination

2.51 J2se 5.0 is required

2.52 Let's see some code!

2.53 Beginning of the class named PlotALot01

2.54 Instantiate two plotting objects

2.55 Feed the plotting object titled "A"

2.56 White random noise

2.57 Plot the data

2.58 Feed and plot the object titled "B"

2.59 The three steps

2.60 Some instance variables

2.61 The first overloaded constructor

2.62 Save the parameter values

2.63 The exception

2.64 A temporary Page object

2.65 Display some information

2.66 Terminate on negative value for sampspacing

2.67 Dispose of the temporary Page object

2.68 Compute and display the remaining plotting parameters

2.69 Terminate on zero traces per page

2.70 Instantiate first usable Page object

2.71 The other overloaded constructor

2.72 The default plotting parameter values

2.73 The feedData method

2.74 Scaling of data to be plotted

2.75 Is the page full?

2.76 Save the data value for plotting later

2.77 The MyCanvas class, a preview

2.78 Call the putdata method to store the data value

2.79 The plotData method

2.80 Two overloaded versions

2.81 Specify the location of the stack

2.82 Make certain that the last page is visible

2.83 Make all pages invisible

2.84 Make the pages visible in reverse order

2.85 Set the location of the stack

2.86 The other overloaded version of the plotData method

2.87 The Page class

2.88 Potentially many page objects...

2.89 The constructor for the page class

2.90 An anonymous terminator for the Page class

2.91 The putData method of the Page class

2.92 The MyCanvas class

2.93 The overridden paint method

2.94 Draw the horizontal axes

2.95 Plot the points

2.96 A vertical offset

2.97 Draw an oval

2.98 Is positive vertical up or down?

2.99 Connect the points with straight lines

2.100 End of the PlotALot01 class

2.101 The class named PlotALot02

2.102 Designed for two-channel data

2.103 Three steps to use the class

2.104 A stack of page objects

2.105 Testing with the main method

2.106 Some data is not random

2.107 The class named PlotALot02 and the main method

2.108 Two data values are required

2.109 The feedData method

2.110 The putData method

2.111 The MyCanvas class

2.112 The overridden paint method

2.113 Setting the drawing color

2.114 New code in the overridden paint method

2.115 End of class plotalot02

2.116 The class named PlotALot03

2.117 Two-channel data on alternating axes

2.118 Three steps for using the class

2.119 A stack of page objects

2.120 Alternating axes

2.121 Testing with the main method

2.122 Some data is not random

2.123 Modified constructor code

2.124 The overridden paint method

2.125 Some code was deleted for brevity

2.126 End of class plotalot03

2.127 The class named PlotALot04

2.128 Three steps for using the class

2.129 A stack of page objects

2.130 Testing with the main method

2.131 Run the programs

2.133 Complete program listings

2.134 Miscellaneous

Fourier analysis

Java1478-fun with java, how and why spectral analysis works

0.1 Table of contents

0.3 Viewing tip

0.5 Multiply-add operations

0.6 Computational requirements for DSP

0.7 Typical notation

0.8 A sum-of-products operation

0.9 An alternative notation

0.11 What is a time series?

0.12 A new time series is produced

0.13 Calculation of the mean or average

0.14 Knowing what to multiply, when, and why

0.15 Multiply-add speed is important

0.16 The net area under the curve

0.17 A periodic example

0.18 The sum of two curves

0.19 The net area under the curve

0.20 Compare Figure 5 To Figure 4

0.21 The net area

0.22 Part of a cycle

0.23 Why is this important?

0.24 The Fourier transform

0.25 Time domain and frequency domain

0.26 Example of time domain and frequency domain

0.27 Forward and inverse transforms

0.28 Sampled time series

0.29 Integration and summation

0.30 The FFT algorithms

0.31 DFT versus FFT

0.32 The DFT algorithm

0.33 What does this really mean?

0.34 It's that simple

0.35 Why does this work?

0.36 Products of sine and cosine functions

0.37 Rewrite and simplify

0.38 What can we learn from these identities?

0.39 Product of two sine functions having the same frequency

0.40 The sum of the product function is not zero

0.41 Product of two cosine functions having the same frequency

0.42 Again the sum of products is not zero

0.43 Product of a sine function and a cosine function

0.44 The sum of the products will be zero

0.45 Neither sine nor cosine

0.46 What about non-matching frequency components?

0.47 Sum and difference frequencies

0.48 A form of measurement error

0.49 Product of two sine functions at different frequencies

0.50 The sum of the product function

0.52 What's next?

0.53 Miscellaneous

1 Java1482-spectrum analysis using java, sampling frequency, folding

1.1 Table of contents

1.3 Viewing tip

1.7 Discussion and sample code

1.8 Sampling frequency and the Nyquist folding frequency

1.9 Display sinusoids

1.10 Input parameters

1.11 The length of the two lists

1.12 Frequency value specifications

1.13 The plotting program named Graph06

1.14 Program output

1.15 Five horizontal plots

1.16 An upper frequency limit

1.17 The Nyquist folding frequency

1.18 The program output

1.19 Not so obvious ...

1.20 Unable to distinguish ...

1.21 One more example

1.22 The program output

1.23 The sinusoids are indistinguishable

1.24 Beginning of the class named Dsp029

1.25 Create array objects to hold sinusoidal data

1.26 Get the parameters

1.27 The getparameters method

1.28 Create the sinusoidal data

1.29 The end of the constructor

1.30 Plotting the sinusoidal data

1.31 Using graph06 to plot the sinusoidal data

1.32 The code for graphintfc01

1.33 What does this do?

1.34 The graphintfc01 interface methods

1.35 The getnmbr method

1.36 The methods named f1, f2, f3, f4, and f5

1.37 One plotting area per method

1.38 How does it work?

1.39 Doesn't know and doesn't care

1.40 Computed "on the fly"

1.41 Returned from an array

1.42 From a disk file, a database, the internet, etc.

1.43 The getNmbr method

1.44 The method named f1

1.45 Round from double to int

1.46 The remaining interface methods

1.47 That's it for dsp029

1.48 Using the interface named graphintfc01

1.49 The program named Graph06

1.50 Operational aspects of graph06

1.51 Spectral Analysis using a DFT Algorithm

1.52 The spectral analysis output

1.53 The format explained

1.54 The frequency axes

1.55 The horizontal units

1.56 Location of the folding frequency

1.57 A peak at the folding frequency

1.58 The frequency folding effect

1.59 Size of the peak at the folding frequency

1.60 A short fat peak at the top

1.61 Mapping the peaks to pixels

1.62 Another DFT example

1.63 Peaks merge at the folding frequency

1.64 The input parameters

1.65 A different data length

1.66 Computational frequency bounds

1.67 Frequencies are close to the folding frequency

1.68 The spectral analysis output

1.69 Peaks are symmetrical about the folding frequency

1.70 Peaks are well defined and wider than before

1.71 Peaks merge at the folding frequency

1.72 Implementing the dft algorithm

1.73 The program named Dsp028

1.74 Computes and displays the magnitude spectrum

1.75 Input parameters

1.76 Specification of sinusoidal frequencies

1.77 Example contents for dsp028.txt

1.78 Performing the spectral analysis

1.79 Will discuss the code in fragments

1.80 Beginning of the class named Dsp028

1.81 Declare array variables

1.82 Always processes five sinusoids

1.83 Create the raw sinusoidal data

1.84 Perform the spectral analysis

1.85 The spectral magnitude results

1.86 The end of the constructor

1.87 The getparameters method

1.88 The interface methods

1.89 The program named Graph03

1.90 The transform method of the ForwardRealToComplex01 class

1.91 Will discuss in fragments

1.92 A brief description

1.93 What does this really mean?

1.94 Compute the complex energy at each frequency

1.95 Nested for loops

1.96 Description of the transform method

1.97 The return values

1.98 The transform method parameters

1.99 Frequency increment, magnitude spectrum, and number of returned values

1.100 The beginning of the transform method

1.101 The remainder of the method and the class

1.102 Store results in output array objects

1.103 Now you know about the dft algorithm

1.104 Spectral analysis using an FFT algorithm

1.105 The program named Dsp030

1.106 Description of the program named dsp030

1.107 The input parameters

1.108 Restrictions on the data length

1.109 The sinusoidal frequency values

1.110 The plotting program

1.111 The transform method

1.112 A sample fft spectral analysis

1.113 Nothing special here

1.114 A matching dft spectral analysis

1.115 Repeat these two experiments

1.116 The ForwardRealToComplexFFT01 class

1.117 Run the programs

1.119 What's next?

1.120 Complete program listings

1.121 Miscellaneous

2 Java1483-spectrum analysis using java, frequency resolution versus data

2.1 Table of contents

2.3 Viewing tip

2.7 Discussion and sample code

2.8 Five pulses in the time domain

2.9 The lengths of the pulses

2.10 Truncated sinusoids

2.11 Will compute at 400 frequencies

2.12 Graph03 and graph06

2.13 The program named Dsp031a

2.14 Time series containing sinusoidal pulses

2.15 The pulse lengths

2.16 Beginning of the class named Dsp031a

2.17 The constructor

2.18 The interface methods

2.19 Spectral analysis results

2.20 Interpretation of the results

2.21 Spectrum of an ideal sinusoid

2.22 Two viewpoints

2.23 The first viewpoint

2.24 The second viewpoint

2.25 A simplified hypothetical situation

2.26 The spectra of the bursts

2.27 The spectral bandwidth of the signal

2.28 The relationship between pulse length and bandwidth

2.29 An expanded plot of the spectral results

2.30 Picking numeric values

2.31 The program named Dsp031

2.32 Description

2.33 Uses a DFT algorithm

2.34 Normalize the results

2.35 Will discuss in fragments

2.36 Beginning of the class named Dsp031

2.37 The constructor

2.38 Perform the spectral analysis

2.39 The interface methods

2.40 What we have learned so far

2.41 Separating closely spaced frequencies

2.42 Five new pulses

2.43 Produced by dsp032a

2.44 Five time series containing pulses

2.45 The sum of two sinusoids

2.46 The frequencies and pulse lengths

2.47 The program named Dsp032a

2.48 Spectral analysis output

2.49 Can we answer the original question now?

2.50 Cannot resolve two peaks for first two pulses

2.51 We must be in luck

2.52 A qualified answer to the question

2.53 What about the effects of noise?

2.54 What can we learn from this?

2.55 Pulse length and frequency separation

2.56 Eureka, we have found it

2.57 The general answer

2.58 The program named Dsp032

2.59 One more experiment

2.60 The five pulses

2.61 Spectral analysis results

2.62 Peaks for first two pulses are now resolvable

2.63 Third pulse hasn't changed

2.64 Spectral peaks for last two pulses are closer

2.65 Expand the horizontal plotting scale

2.66 The peaks are barely resolvable

2.67 The program named Dsp033

2.68 The spectral analysis

2.69 Computation of the frequencies

2.70 Run the programs

2.72 What's next?

2.73 Complete program listings

2.74 Miscellaneous

3 Java1484-spectrum analysis using java, complex spectrum and phase angle

3.1 Table of contents

3.3 Viewing tip

3.7 Discussion and sample code

3.8 The order of the plotted results

3.9 Parameter file format

3.10 Format for frequency specifications

3.11 Sample parameters for testing

3.12 The plotting programs

3.13 Spectral analysis

3.15 The input pulse

3.16 Potential energy in the spring

3.17 Kinetic energy in the mass

3.18 An exchange of energy

3.19 Capturing the position of the mass as a sampled time series

3.20 Computing the Fourier transform

3.21 The results of performing a spectral analysis on the pulse

3.22 The amplitude spectrum

3.23 The real and imaginary parts of the transform

3.24 The phase angle in degrees

3.25 How is the phase angle computed?

3.26 How should we interpret the phase angle?

3.27 Many combinations

3.28 The frequency band of primary interest

3.29 A nonlinear phase angle, so what?

3.30 Computation of the phase angle

3.31 What if both values are zero?

3.32 The atan method of the math class

3.33 Correction required for the quadrant

3.34 The first and third quadrants

3.35 The second and fourth quadrants

3.36 An exercise for the reader

3.37 The program named Dsp034

3.38 The beginning of the Dsp034 class

3.39 The damped sinusoidal pulse

3.40 Store the pulse in the time series

3.41 Print the parameters and the pulse

3.42 Perform the spectral analysis

3.43 The getparameters method

3.44 The interface methods

3.45 More examples

3.46 The simplest pulse of all, an impulse

3.47 Screen output from the getparameters method

3.48 The length of the pulse

3.49 Defining the origin of time

3.50 Computational frequency range

3.51 The values that make up the pulse

3.52 Results of spectral analysis on an impulse

3.53 Where is that impulse?

3.54 The amplitude spectrum

3.55 The real spectrum

3.56 The imaginary spectrum

3.57 The phase angle

3.58 Introduce a time delay

3.59 The spectral analysis output

3.60 The amplitude spectrum

3.61 The real and imaginary parts of the spectrum

3.62 Real and imaginary values are intrinsically linked

3.63 The phase angle spectrum

3.64 An important conclusion

3.65 The converse is also true

3.66 A more acceptable form of phase distortion

3.67 Introduce a large time delay

3.68 The output from the spectral analysis

3.69 A boxcar pulse

3.70 The spectral analysis output for the boxcar pulse

3.71 The amplitude spectrum

3.72 The magic of non real-time digital processing

3.73 The spectral analysis output

3.74 Probably more than you ever wanted to know

3.75 Run the program

3.77 What's next?

3.78 Complete program listing

3.79 Miscellaneous

4 Java1485-spectrum analysis using java, forward and inverse transforms,

4.1 Table of contents

4.3 Viewing tip

4.7 Discussion and sample code

4.8 Description of the program named Dsp035

4.9 The order of the plotted results

4.10 The format of the plots

4.11 The forward Fourier transform

4.12 The inverse Fourier transform

4.14 Length is a power of two

4.15 The input time series

4.16 The objective

4.17 The real part of the spectrum is symmetrical

4.18 The imaginary part of the spectrum is asymmetrical

4.19 The converse is also true

4.20 The amplitude spectrum

4.21 The output time series

4.22 The program named Dsp035

4.23 The beginning of the class named Dsp035

4.24 Beginning of the constructor

4.25 Compute the complex spectrum

4.26 The method parameters

4.27 Perform the inverse Fourier transform

4.28 An object of the class dsp035

4.29 The interface methods

4.30 The InverseComplexToReal01 class

4.31 Parameters for the inverseTransform method

4.32 Assumptions

4.33 A real output

4.34 Beginning of the inverseTransform method

4.35 The program named Dsp036

4.36 The output from Dsp036

4.37 Some code from Dsp036

4.38 The forward fourier transform

4.39 The inverse fourier transform

4.40 A couple of things to note

4.41 The signature of the complexToComplex method

4.42 The first parameter of the complextocomplex method

4.43 The forward transform

4.44 The inverse transform

4.45 Fft and dft produce equivalent results

4.46 Using a Fourier transform to perform frequency filtering

4.47 Operation of the program

4.48 Modify the complex spectrum

4.49 The plotting format

4.50 Perform an inverse fourier transform

4.51 The filtered impulse

4.52 Beginning of the class for Dsp037

4.53 Beginning of the constructor

4.54 Compute the Fourier transform

4.55 Apply the filter to the frequency data

4.56 Re-compute the magnitude

4.57 Compute the inverse Fourier transform

4.58 Display the results

4.59 One more example, Dsp038

4.60 Compare the results

4.61 Compare Figure 5 With Figure 4

4.62 Compare the bandwidth of the pass band

4.63 Waveforms of filtered impulse

4.64 Run the programs

4.66 Complete program listings

4.67 Miscellaneous

5 Java1486-fun with java, understanding the fast fourier transform (fft)

5.1 Table of contents

5.3 Viewing tip

5.6 General discussion

5.7 A general-purpose transform

5.8 Transforming from space domain to wave number domain

5.9 Two-dimensional Fourier transforms

5.10 Wave-number response to seismic waves

5.11 A general purpose mathematical transform

5.12 Fourier transform images

5.13 Will discuss underlying concepts

5.14 A linear transform

5.15 Output display of the FFT applet

5.16 Real and imaginary sections

5.17 A complex sample

5.18 Powers of two

5.19 Location of the origin

5.20 Other applet controls

5.21 Back to the concept of the linear transform

5.22 A mirror-image pulse

5.23 The transform output

5.24 Now add the two input series

5.25 The transform of the sum equals the sum of the transforms

5.26 Normalized output

5.27 The real part is the same

5.28 The imaginary part sums to zero

5.29 Single sample real pulse (impulse) with a delay

5.30 A shift of zero

5.31 A shift of one sample interval

5.32 A cosine curve and a sine curve

5.33 Number of output samples equals number of input samples

5.34 Representing time and frequency

5.35 A shift of two sample intervals

5.36 A shift of four sample intervals

5.37 Equations to describe the real and imaginary parts of the transform

5.38 Transformation of a complex series

5.39 Linearity still applies

5.40 Can produce the transform of a time series by the adding transforms of

5.41 The input series is the sum of the individual samples

5.42 Steps in the FFT algorithm

5.43 A sample program

5.44 Separate processes in an FFT algorithm

5.45 How the processes are implemented

5.46 Three cases are examined

5.47 The program named Fft02

5.48 Instantiate a Transform object

5.49 The class named Transform

5.50 The method parameters

5.51 Performing the transform

5.52 The transforms of the complex input samples

5.53 Correct for actual position and recombine

5.54 The correctAndRecombine method

5.55 Back to the main method

5.56 The graphic form of Case A

5.57 The numeric output for Case A

5.58 Case B code

5.59 Case B in graphical form

5.60 Case B output in numeric form

5.61 Case C code

5.62 The graphic form of Case C

5.63 Case C output in numeric form

5.64 The display method

5.65 Run the program

5.67 Complete program listings

5.68 Miscellaneous

6 Java1490-2d fourier transforms using java, part 1

6.1 Table of contents

6.3 Two separate programs

6.4 Digital signal processing (DSP)

6.5 Will use in subsequent modules

6.6 Viewing tip

6.8 General discussion

6.9 Time domain and frequency domain

6.10 A one-dimensional Fourier transform

6.11 Time domain data is purely real

6.12 The space domain

6.13 Time and space are analogous

6.14 Frequency and wavenumber are analogous

6.15 Period and wavelength are analogous

6.16 Some real world examples

6.17 A commercial radio station

6.18 Where people live

6.19 A limit on the transmitting power

6.20 A directional transmitting antenna

6.21 The design of the antenna system

6.22 Basic concepts

6.23 A one-dimensional space

6.24 Standing waves

6.25 What does a standing wave look like?

6.26 Add the sensor output signals

6.27 A one-half wavelength spacing

6.28 Change the wavelength of the standing wave

6.29 A two-element array

6.30 How can we compute the array response?

6.31 A sampled space series

6.32 Three example wavenumber spectra

6.33 Plots of 3D surfaces

6.34 Black, white, and the colors in between

6.35 The maximum array output

6.36 Response of the leftmost array

6.37 The Nyquist folding wavenumber

6.38 Estimating the array response from the colors

6.39 Additional separation between array elements

6.40 Separate the elements even more

6.41 A wavenumber filter

6.42 A two-element array is fairly limiting

6.43 Let's apply some weights

6.44 The leftmost pair of images

6.45 The wavenumber response

6.46 Could continue adding elements to the array

6.47 A more selective wavenumber filter

6.48 A weighted three-element array

6.49 A three-element array with negative weighting

6.50 The wavenumber response

6.51 Two full peaks but at different locations

6.52 What can we learn from these scenarios?

6.53 Extending into two dimensions

6.54 Move the array to a table top

6.55 What does a wave look like in this scenario?

6.56 The angle of attack is important

6.57 Infinite wavelength

6.58 The 2D wavenumber response of a linear array

6.59 The response for a constant wavenumber

6.60 Response versus direction

6.62 Not good for the radio transmitter

6.63 A two-dimensional array

6.64 Computing the wavenumber response

6.65 A constant wave number

6.66 Assume a wavenumber

6.67 Determine the response versus direction

6.68 A possible solution to the problem

6.69 Arrays are used in various applications

6.70 Radio astronomy

6.71 Seismology

6.74 Petroleum exploration

6.75 Image processing

6.76 Image processing in the wavenumber domain

6.77 Image processing in the space domain

6.78 Convolution versus multiplication

6.79 A simple space function and its wavenumber spectrum

6.80 A 2D convolution operator and its spectral response

6.81 The result of convolution

6.82 Positive and negative fences

6.83 Edge detection

6.84 Wavenumber spectrum of the convolution output

6.85 Hidden watermarks and trademarks

6.87 What's next?

6.88 Miscellaneous

7 Java1491-2d fourier transforms using java, part 2

7.1 Table of contents

7.3 Two separate programs

7.4 Using the class named ImgMod30

7.5 Digital signal processing (DSP)

7.6 Will use in subsequent modules

7.7 Viewing tip

7.10 General discussion

7.11 The space domain

7.12 A purely real space domain

7.14 Sample programs

7.15 The class named ImgMod30

7.16 The xform2D method

7.17 Parameters

7.18 The remainder of the xform2D method

7.19 The inverseXform2D method

7.20 Parameters

7.21 The shiftOrigin method

7.22 Two views of the same wavenumber spectrum

7.23 How the wavenumber spectrum is actually computed

7.24 Computationally sound but not visually pleasing

7.25 The wavenumber spectrum is periodic

7.26 The shiftOrigin method code

7.27 End of the ImgMod30 class

7.28 The program named ImgMod31

7.29 Command line parameters

7.30 Forward and inverse Fourier transforms

7.31 Fourteen cases

7.32 A stack of output images

7.33 Numeric output

7.34 The size of the surfaces

7.35 The quality of the transform process

7.36 Matching pairs of values

7.37 How to use the program named ImgMod31

7.38 Program code for ImgMod31

7.39 Create and save the test surface

7.40 Display the test surface

7.41 An example test surface plot

7.42 Perform the forward Fourier transform

7.43 Prepare array objects to receive the transform results

7.44 Perform the forward transform

7.45 Display unshifted amplitude spectrum

7.46 Need to shift the origin for display

7.47 Shift the origin and display the results

7.48 Example displays

7.49 Perform an inverse transform

7.50 Prepare an array object to store the results

7.51 Call the inverseXform2D method

7.52 Display the result of the inverse transform

7.53 Display some numeric results

7.54 What do we know so far?

7.55 The getSpatialData method

7.56 The code for the getSpatialData method

7.57 Code and graphic output for fourteen example cases

7.59 The output surface

7.60 The wavenumber spectrum is flat

7.61 A very small error

7.63 The code for Case 1

7.64 The visual output

7.65 The input and output surfaces match visually

7.66 The real and imaginary parts

7.68 The amplitude spectrum is ugly

7.69 A very small error

7.71 A box on the diagonal in space

7.72 What do we know?

7.73 The code for Case 2

7.74 The real and imaginary parts of the spectrum

7.75 A different plotting color scheme

7.76 The amplitude spectrum

7.77 The output surface

7.78 The overall results

7.80 Amplitude spectrum should not change

7.81 The real and imaginary parts of the spectrum may change

7.82 A hypothetical example

7.83 Amplitude spectrum is the same

7.85 Another example of sin(x)/x

7.86 Our expectations are borne out

7.88 Rotated by ninety degrees

7.89 A wider peak

7.90 The real and imaginary parts

7.91 The final output matches the input

7.93 Many weighted lines of points

7.94 Black is -1, white is +1

7.95 Symmetries on four different axes

7.96 Must be zero at the wavenumber origin

7.97 Peaks at the folding wave numbers

7.98 The real and imaginary parts

7.99 The final output

7.101 Need to change the surface plotting scale

7.102 Display fewer results

7.103 A zero frequency sine wave

7.104 The code for this case

7.105 A single point at the origin

7.106 The inverse transform output is ugly

7.107 A very small error

7.109 The graphic output for Case 8

7.111 The wavenumber spectrum

7.112 The inverse Fourier transform output

7.115 The wavenumber spectrum

7.116 The inverse Fourier transform output

7.118 The wavenumber spectrum

7.119 Inverse Fourier transform output

7.121 The wavenumber spectrum

7.122 The inverse Fourier transform output

7.123 The end of the getSpatialData method

7.124 Run the program

7.126 Complete program listings

7.127 Miscellaneous

Convolution and frequency filtering

Java1487-convolution and frequency filtering in java

1 Java1488-convolution and matched filtering in java

Adaptive processing

Java2350-adaptive filtering in java, getting started

1 Java2352-an adaptive whitening filter in java

2 Java2354-a general-purpose lms adaptive engine in java

3 Java2356-an adaptive line tracker in java

3.1 Table of contents

3.3 Tutorial Links

3.4 Legacy content and references

3.5 Legacy versus openstax presentation format

3.6 Miscellaneous

4 Java2358-adaptive identification and inverse filtering using java

5 Java2360-adaptive noise cancellation using java

6 Java2362-adaptive prediction using java

Data compression and the discrete cosine transform

Java2440 understanding the lempel-ziv data compression algorithm in

0.1 Table of contents

0.3 Tutorial and code links

0.4 Miscellaneous

1 Java2442 understanding the huffman data compression algorithm in java

2 Java2444 understanding the discrete cosine transform in java

3 Java2446 understanding the 2d discrete cosine transform in java

4 Java2448 understanding the 2d discrete cosine transform in java, part

Recursive filtering

Source: OpenStax, Digital signal processing - dsp. OpenStax CNX. Jan 06, 2016 Download for free at https://legacy.cnx.org/content/col11642/1.38

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