Fourier analysis

OpenStax

@

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

1 Table of contents

5 Multiply-add operations

6 Computational requirements for DSP

7 Typical notation

8 A sum-of-products operation

9 An alternative notation

11 What is a time series?

12 A new time series is produced

13 Calculation of the mean or average

14 Knowing what to multiply, when, and why

15 Multiply-add speed is important

16 The net area under the curve

17 A periodic example

18 The sum of two curves

19 The net area under the curve

20 Compare Figure 5 To Figure 4

21 The net area

22 Part of a cycle

23 Why is this important?

24 The Fourier transform

25 Time domain and frequency domain

26 Example of time domain and frequency domain

27 Forward and inverse transforms

28 Sampled time series

29 Integration and summation

30 The FFT algorithms

31 DFT versus FFT

32 The DFT algorithm

33 What does this really mean?

34 It's that simple

35 Why does this work?

36 Products of sine and cosine functions

37 Rewrite and simplify

38 What can we learn from these identities?

39 Product of two sine functions having the same frequency

40 The sum of the product function is not zero

41 Product of two cosine functions having the same frequency

42 Again the sum of products is not zero

43 Product of a sine function and a cosine function

44 The sum of the products will be zero

45 Neither sine nor cosine

46 What about non-matching frequency components?

47 Sum and difference frequencies

48 A form of measurement error

49 Product of two sine functions at different frequencies

50 The sum of the product function

52 What's next?

53 Miscellaneous

Java1482-spectrum analysis using java, sampling frequency, folding

1 Table of contents

7 Discussion and sample code

8 Sampling frequency and the Nyquist folding frequency

9 Display sinusoids

10 Input parameters

11 The length of the two lists

12 Frequency value specifications

13 The plotting program named Graph06

14 Program output

15 Five horizontal plots

16 An upper frequency limit

17 The Nyquist folding frequency

18 The program output

19 Not so obvious ...

20 Unable to distinguish ...

21 One more example

22 The program output

23 The sinusoids are indistinguishable

24 Beginning of the class named Dsp029

25 Create array objects to hold sinusoidal data

26 Get the parameters

27 The getparameters method

28 Create the sinusoidal data

29 The end of the constructor

30 Plotting the sinusoidal data

31 Using graph06 to plot the sinusoidal data

32 The code for graphintfc01

33 What does this do?

34 The graphintfc01 interface methods

35 The getnmbr method

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

37 One plotting area per method

38 How does it work?

39 Doesn't know and doesn't care

40 Computed "on the fly"

41 Returned from an array

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

43 The getNmbr method

44 The method named f1

45 Round from double to int

46 The remaining interface methods

47 That's it for dsp029

48 Using the interface named graphintfc01

49 The program named Graph06

50 Operational aspects of graph06

51 Spectral Analysis using a DFT Algorithm

52 The spectral analysis output

53 The format explained

54 The frequency axes

55 The horizontal units

56 Location of the folding frequency

57 A peak at the folding frequency

58 The frequency folding effect

59 Size of the peak at the folding frequency

60 A short fat peak at the top

61 Mapping the peaks to pixels

62 Another DFT example

63 Peaks merge at the folding frequency

64 The input parameters

65 A different data length

66 Computational frequency bounds

67 Frequencies are close to the folding frequency

68 The spectral analysis output

69 Peaks are symmetrical about the folding frequency

70 Peaks are well defined and wider than before

71 Peaks merge at the folding frequency

72 Implementing the dft algorithm

73 The program named Dsp028

74 Computes and displays the magnitude spectrum

75 Input parameters

76 Specification of sinusoidal frequencies

77 Example contents for dsp028.txt

78 Performing the spectral analysis

79 Will discuss the code in fragments

80 Beginning of the class named Dsp028

81 Declare array variables

82 Always processes five sinusoids

83 Create the raw sinusoidal data

84 Perform the spectral analysis

85 The spectral magnitude results

86 The end of the constructor

87 The getparameters method

88 The interface methods

89 The program named Graph03

90 The transform method of the ForwardRealToComplex01 class

91 Will discuss in fragments

92 A brief description

93 What does this really mean?

94 Compute the complex energy at each frequency

95 Nested for loops

96 Description of the transform method

97 The return values

98 The transform method parameters

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

100 The beginning of the transform method

101 The remainder of the method and the class

102 Store results in output array objects

103 Now you know about the dft algorithm

104 Spectral analysis using an FFT algorithm

105 The program named Dsp030

106 Description of the program named dsp030

107 The input parameters

108 Restrictions on the data length

109 The sinusoidal frequency values

110 The plotting program

111 The transform method

112 A sample fft spectral analysis

113 Nothing special here

114 A matching dft spectral analysis

115 Repeat these two experiments

116 The ForwardRealToComplexFFT01 class

117 Run the programs

119 What's next?

120 Complete program listings

121 Miscellaneous

Java1483-spectrum analysis using java, frequency resolution versus data

1 Table of contents

7 Discussion and sample code

8 Five pulses in the time domain

9 The lengths of the pulses

10 Truncated sinusoids

11 Will compute at 400 frequencies

12 Graph03 and graph06

13 The program named Dsp031a

14 Time series containing sinusoidal pulses

15 The pulse lengths

16 Beginning of the class named Dsp031a

17 The constructor

18 The interface methods

19 Spectral analysis results

20 Interpretation of the results

21 Spectrum of an ideal sinusoid

22 Two viewpoints

23 The first viewpoint

24 The second viewpoint

25 A simplified hypothetical situation

26 The spectra of the bursts

27 The spectral bandwidth of the signal

28 The relationship between pulse length and bandwidth

29 An expanded plot of the spectral results

30 Picking numeric values

31 The program named Dsp031

33 Uses a DFT algorithm

34 Normalize the results

35 Will discuss in fragments

36 Beginning of the class named Dsp031

37 The constructor

38 Perform the spectral analysis

39 The interface methods

40 What we have learned so far

41 Separating closely spaced frequencies

42 Five new pulses

43 Produced by dsp032a

44 Five time series containing pulses

45 The sum of two sinusoids

46 The frequencies and pulse lengths

47 The program named Dsp032a

48 Spectral analysis output

49 Can we answer the original question now?

50 Cannot resolve two peaks for first two pulses

51 We must be in luck

52 A qualified answer to the question

53 What about the effects of noise?

54 What can we learn from this?

55 Pulse length and frequency separation

56 Eureka, we have found it

57 The general answer

58 The program named Dsp032

59 One more experiment

60 The five pulses

61 Spectral analysis results

62 Peaks for first two pulses are now resolvable

63 Third pulse hasn't changed

64 Spectral peaks for last two pulses are closer

65 Expand the horizontal plotting scale

66 The peaks are barely resolvable

67 The program named Dsp033

68 The spectral analysis

69 Computation of the frequencies

70 Run the programs

72 What's next?

73 Complete program listings

74 Miscellaneous

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

1 Table of contents

7 Discussion and sample code

8 The order of the plotted results

9 Parameter file format

10 Format for frequency specifications

11 Sample parameters for testing

12 The plotting programs

13 Spectral analysis

15 The input pulse

16 Potential energy in the spring

17 Kinetic energy in the mass

18 An exchange of energy

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

20 Computing the Fourier transform

21 The results of performing a spectral analysis on the pulse

22 The amplitude spectrum

23 The real and imaginary parts of the transform

24 The phase angle in degrees

25 How is the phase angle computed?

26 How should we interpret the phase angle?

27 Many combinations

28 The frequency band of primary interest

29 A nonlinear phase angle, so what?

30 Computation of the phase angle

31 What if both values are zero?

32 The atan method of the math class

33 Correction required for the quadrant

34 The first and third quadrants

35 The second and fourth quadrants

36 An exercise for the reader

37 The program named Dsp034

38 The beginning of the Dsp034 class

39 The damped sinusoidal pulse

40 Store the pulse in the time series

41 Print the parameters and the pulse

42 Perform the spectral analysis

43 The getparameters method

44 The interface methods

45 More examples

46 The simplest pulse of all, an impulse

47 Screen output from the getparameters method

48 The length of the pulse

49 Defining the origin of time

50 Computational frequency range

51 The values that make up the pulse

52 Results of spectral analysis on an impulse

53 Where is that impulse?

54 The amplitude spectrum

55 The real spectrum

56 The imaginary spectrum

57 The phase angle

58 Introduce a time delay

59 The spectral analysis output

60 The amplitude spectrum

61 The real and imaginary parts of the spectrum

62 Real and imaginary values are intrinsically linked

63 The phase angle spectrum

64 An important conclusion

65 The converse is also true

66 A more acceptable form of phase distortion

67 Introduce a large time delay

68 The output from the spectral analysis

69 A boxcar pulse

70 The spectral analysis output for the boxcar pulse

71 The amplitude spectrum

72 The magic of non real-time digital processing

73 The spectral analysis output

74 Probably more than you ever wanted to know

75 Run the program

77 What's next?

78 Complete program listing

79 Miscellaneous

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

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

1 Table of contents

6 General discussion

7 A general-purpose transform

8 Transforming from space domain to wave number domain

9 Two-dimensional Fourier transforms

10 Wave-number response to seismic waves

11 A general purpose mathematical transform

12 Fourier transform images

13 Will discuss underlying concepts

14 A linear transform

15 Output display of the FFT applet

16 Real and imaginary sections

17 A complex sample

18 Powers of two

19 Location of the origin

20 Other applet controls

21 Back to the concept of the linear transform

22 A mirror-image pulse

23 The transform output

24 Now add the two input series

25 The transform of the sum equals the sum of the transforms

26 Normalized output

27 The real part is the same

28 The imaginary part sums to zero

29 Single sample real pulse (impulse) with a delay

30 A shift of zero

31 A shift of one sample interval

32 A cosine curve and a sine curve

33 Number of output samples equals number of input samples

34 Representing time and frequency

35 A shift of two sample intervals

36 A shift of four sample intervals

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

38 Transformation of a complex series

39 Linearity still applies

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

41 The input series is the sum of the individual samples

42 Steps in the FFT algorithm

43 A sample program

44 Separate processes in an FFT algorithm

45 How the processes are implemented

46 Three cases are examined

47 The program named Fft02

48 Instantiate a Transform object

49 The class named Transform

50 The method parameters

51 Performing the transform

52 The transforms of the complex input samples

53 Correct for actual position and recombine

54 The correctAndRecombine method

55 Back to the main method

56 The graphic form of Case A

57 The numeric output for Case A

59 Case B in graphical form

60 Case B output in numeric form

62 The graphic form of Case C

63 Case C output in numeric form

64 The display method

65 Run the program

67 Complete program listings

68 Miscellaneous

Java1490-2d fourier transforms using java, part 1

1 Table of contents

3 Two separate programs

4 Digital signal processing (DSP)

5 Will use in subsequent modules

8 General discussion

9 Time domain and frequency domain

10 A one-dimensional Fourier transform

11 Time domain data is purely real

12 The space domain

13 Time and space are analogous

14 Frequency and wavenumber are analogous

15 Period and wavelength are analogous

16 Some real world examples

17 A commercial radio station

18 Where people live

19 A limit on the transmitting power

20 A directional transmitting antenna

21 The design of the antenna system

22 Basic concepts

23 A one-dimensional space

24 Standing waves

25 What does a standing wave look like?

26 Add the sensor output signals

27 A one-half wavelength spacing

28 Change the wavelength of the standing wave

29 A two-element array

30 How can we compute the array response?

31 A sampled space series

32 Three example wavenumber spectra

33 Plots of 3D surfaces

34 Black, white, and the colors in between

35 The maximum array output

36 Response of the leftmost array

37 The Nyquist folding wavenumber

38 Estimating the array response from the colors

39 Additional separation between array elements

40 Separate the elements even more

41 A wavenumber filter

42 A two-element array is fairly limiting

43 Let's apply some weights

44 The leftmost pair of images

45 The wavenumber response

46 Could continue adding elements to the array

47 A more selective wavenumber filter

48 A weighted three-element array

49 A three-element array with negative weighting

50 The wavenumber response

51 Two full peaks but at different locations

52 What can we learn from these scenarios?

53 Extending into two dimensions

54 Move the array to a table top

55 What does a wave look like in this scenario?

56 The angle of attack is important

57 Infinite wavelength

58 The 2D wavenumber response of a linear array

59 The response for a constant wavenumber

60 Response versus direction

62 Not good for the radio transmitter

63 A two-dimensional array

64 Computing the wavenumber response

65 A constant wave number

66 Assume a wavenumber

67 Determine the response versus direction

68 A possible solution to the problem

69 Arrays are used in various applications

70 Radio astronomy

74 Petroleum exploration

75 Image processing

76 Image processing in the wavenumber domain

77 Image processing in the space domain

78 Convolution versus multiplication

79 A simple space function and its wavenumber spectrum

80 A 2D convolution operator and its spectral response

81 The result of convolution

82 Positive and negative fences

83 Edge detection

84 Wavenumber spectrum of the convolution output

85 Hidden watermarks and trademarks

87 What's next?

88 Miscellaneous

Java1491-2d fourier transforms using java, part 2

1 Table of contents

3 Two separate programs

4 Using the class named ImgMod30

5 Digital signal processing (DSP)

6 Will use in subsequent modules

10 General discussion

11 The space domain

12 A purely real space domain

14 Sample programs

15 The class named ImgMod30

16 The xform2D method

18 The remainder of the xform2D method

19 The inverseXform2D method

21 The shiftOrigin method

22 Two views of the same wavenumber spectrum

23 How the wavenumber spectrum is actually computed

24 Computationally sound but not visually pleasing

25 The wavenumber spectrum is periodic

26 The shiftOrigin method code

27 End of the ImgMod30 class

28 The program named ImgMod31

29 Command line parameters

30 Forward and inverse Fourier transforms

31 Fourteen cases

32 A stack of output images

33 Numeric output

34 The size of the surfaces

35 The quality of the transform process

36 Matching pairs of values

37 How to use the program named ImgMod31

38 Program code for ImgMod31

39 Create and save the test surface

40 Display the test surface

41 An example test surface plot

42 Perform the forward Fourier transform

43 Prepare array objects to receive the transform results

44 Perform the forward transform

45 Display unshifted amplitude spectrum

46 Need to shift the origin for display

47 Shift the origin and display the results

48 Example displays

49 Perform an inverse transform

50 Prepare an array object to store the results

51 Call the inverseXform2D method

52 Display the result of the inverse transform

53 Display some numeric results

54 What do we know so far?

55 The getSpatialData method

56 The code for the getSpatialData method

57 Code and graphic output for fourteen example cases

59 The output surface

60 The wavenumber spectrum is flat

61 A very small error

63 The code for Case 1

64 The visual output

65 The input and output surfaces match visually

66 The real and imaginary parts

68 The amplitude spectrum is ugly

69 A very small error

71 A box on the diagonal in space

72 What do we know?

73 The code for Case 2

74 The real and imaginary parts of the spectrum

75 A different plotting color scheme

76 The amplitude spectrum

77 The output surface

78 The overall results

80 Amplitude spectrum should not change

81 The real and imaginary parts of the spectrum may change

82 A hypothetical example

83 Amplitude spectrum is the same

85 Another example of sin(x)/x

86 Our expectations are borne out

88 Rotated by ninety degrees

89 A wider peak

90 The real and imaginary parts

91 The final output matches the input

93 Many weighted lines of points

94 Black is -1, white is +1

95 Symmetries on four different axes

96 Must be zero at the wavenumber origin

97 Peaks at the folding wave numbers

98 The real and imaginary parts

99 The final output

101 Need to change the surface plotting scale

102 Display fewer results

103 A zero frequency sine wave

104 The code for this case

105 A single point at the origin

106 The inverse transform output is ugly

107 A very small error

109 The graphic output for Case 8

111 The wavenumber spectrum

112 The inverse Fourier transform output

115 The wavenumber spectrum

116 The inverse Fourier transform output

118 The wavenumber spectrum

119 Inverse Fourier transform output

121 The wavenumber spectrum

122 The inverse Fourier transform output

123 The end of the getSpatialData method

124 Run the program

126 Complete program listings

127 Miscellaneous

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

	4 Gang of Four Design Patterns By JavaChamp Team Start Quiz
	19 Sociology 19 Health and Medicine MCQ By OpenStax Start Quiz
	3 BOD GI quiz By Brooke Delaney Start Exam
	1 Psychology Concept Test By John Gabrieli Start Test
	1 Business Law MCQ 1 By Maureen Miller Start Exam
©flickr: Luis	Final Exam Review By Madison Christian Start Exam
	15 Biology 15 Genes and Proteins MCQ By OpenStax Start Quiz
	25 Biology 25 Seedless Plants MCQ By OpenStax Start Quiz
	9 BOD- Liver Quiz .... By Brooke Delaney Start Quiz
	2011 Dynamics CRM By Danielrosenberger Start Quiz