Basic Properties of signal, Fourier Expansion and it’s Applications in Digital Image processing.

1. Basic Properties of signal, Fourier Expansion and it’s Applications in Digital Image processing. Md. Al Mehedi Hasan Assistant Professor Dept. of Computer Science & Engineering RUET, Rajshahi-6204. E-mail: mehedi_ru@yahoo.com

2. Signals • Signal is defined by its Amplitude, Frequency and Phase • Signals can be analog or digital. • Analog signals can have an infinite number of values in a range. • Digital signals can have only a limited number of values.

3. Comparison of analog and digital signals

4. Periodic Signal Both analog and digital signals can be of two forms: Periodic and Aperiod. A signal is a periodic if it completes a pattern within a measurable time frame, called a period, and repeats that pattern over identical subsequent period.

5. Periodic signals (continue) Periodic signals can be classified as simple or composite. simple composite

7. Aperiodic Signal An aperiodic, or nonperiodic, signal has no patterns.

8. Amplitude The Amplitude of a signal is the value of the signal at any point on the wave.

9. Period and Frequency Period refers to the amount of time, a signal needs to complete one cycle. Frequency refers to the number of periods in one second. Frequency and period are the inverse of each other.

10. Units of period and frequency

11. Phase The term phase describes the position of the waveform relative to time zero.

12. Two signals with the same phase and frequency, but different amplitudes

13. Two signals with the same amplitude and phase, but different frequencies

14. Three sine waves with the same amplitude and frequency, but different phases

15. Example The power we use at home has a frequency of 60 Hz. The period of this sine wave can be determined as follows:

16. Example The period of a signal is 100 ms. What is its frequency in kilohertz? Solution First we change 100 ms to seconds, and then we calculate the frequency from the period (1 Hz = 10−3 kHz).

17. Note If a signal does not change at all, its frequency is zero. If a signal changes instantaneously, its frequency is infinite.

18. Example A sine wave is offset 1/6 cycle with respect to time 0. What is its phase in degrees and radians? Solution We know that 1 complete cycle is 360°. Therefore, 1/6 cycle is

19. Sine and Cosine Functions • Periodic functions • General form of sine and cosine functions:

20. Sine and Cosine Functions Special case: A=1, b=0, α=1 π

21. Sine and Cosine Functions (cont’d) • Shifting or translating the sine function by a const b Cosine is a shifted sine function:

22. Sine and Cosine Functions (cont’d) • Changing the amplitude A

23. Sine and Cosine Functions (cont’d) • Changing the period T=2π/|α| • e.g., y=cos(αt) α =4 period 2π/4=π/2 shorter period higher frequency (i.e., oscillates faster) Frequency is defined as f=1/T Different notation: sin(αt)=sin(2πt/T)=sin(2πft)

24. Definition of Radian One radian is the measure of a central angle that intercepts an arc equal in length to the radius of the circle. See Figure. Algebraically, this means that where θ is measured in radians.

28. Why Radian Measure? In most applications of trigonometry, angles are measured in degrees. In more advances work in mathematics, radian measure of angles is preferred. Radian measure allows us to treat the trigonometric functions as functions with domains of real numbers, rather than angles.

29. Linear speed measures how fast the particle moves, and angular speed measures how fast the angle changes.

32. Frequency and Angular Frequency Frequency is a metric for expressing the rate of oscillation in a wave. For planar and longitudinal waves, this often expressed in oscillations-per-second or Hz. Angular frequency used for expressing rates of rotation, similar to revolutions-per-second, and is usually expressed in radians-per-second. It can be thought of as a wave with a constant amplitude where the amplitude rotates in a circle in space.

33. Different Notation of Sine and Cosine Functions • Changing the period T=2π/|α| • e.g., y=cos(αt) α =4 period 2π/4=π/2 shorter period higher frequency (i.e., oscillates faster) Frequency is defined as f=1/T

34. Different Notation of Sine and Cosine Functions (continue) Fundamental Frequency?

35. Time Domain and Frequency Domain Time Domain: The time-domain plot shows changes in signal amplitude with respect to time. Phase and frequency are not explicitly measure on a time-domain plot. Frequency Domain: The time-domain plot shows changes in signal amplitude with respect to frequency.

36. The time-domain and frequency-domain plots of a sine wave

37. The time domain and frequency domain of three sine waves

38. Frequency Spectrum and Bandwidth The frequency spectrum of a signal is the combination of all sine wave signals that make up that signal. The bandwidth of a signal is the width of the frequency spectrum.

39. Jean Baptiste Joseph Fourier Fourier was born in Auxerre, France in 1768 • Most famous for his work “La Théorie Analitique de la Chaleur” published in 1822 • Translated into English in 1878: “The Analytic Theory of Heat” Nobody paid much attention when the work was first published One of the most important mathematical theories in modern engineering

40. Images taken from Gonzalez & Woods, Digital Image Processing (2002) The Big Idea = Any function that periodically repeats itself can be expressed as a sum of sines and cosines of different frequencies each multiplied by a different coefficient – a Fourier series

41. Fourier analysis • A single-frequency sine wave is not useful in some situation • We need to use a composite signal, a signal made of many simple sine waves. • According to Fourier analysis, any composite signal is a combination of simple sine waves with different frequencies, amplitudes, and phases.

42. Composite Signals and Periodicity • If the composite signal is periodic, the decomposition gives a series of signals with discrete frequencies. • If the composite signal is nonperiodic, the decomposition gives a combination of sine waves with continuous frequencies.

43. Fourier Series of composite periodic signal • Every composite periodic signal can be represented with a series of sine and cosine functions. • The functions are integral harmonics of the fundamental frequency “f” of the composite signal. • Using the series we can decompose any periodic signal into its harmonics.

44. A composite periodic signal

45. Decomposition of a composite periodic signal in the time and frequency domains

46. Nonperiodic signal The time and frequency domains of a nonperiodic signal

47. Fourier Series

48. Fourier Series Where

49. Meaning of Coefficients

50. An equation with many faces There are several different ways to write the Fourier series.