1. README Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain the slide with highlighted notes.

2. STFT as Filter Bank Introduction to Wavelet Transform Yen-Ming Lai Doo-hyun Sung November 15, 2010 ENEE630, Project 1

3. Wavelet Tutorial Overview DFT as filter bank STFT as filter bank Wavelet transform as filter bank

4. Discrete Fourier Transform

5. DFT for fixed w_0

6. fix specific frequency w_0

7. DFT for fixed w_0 pass in input signal x(n)

8. DFT for fixed w_0 modulate by complex exponential of frequency w_0

9. DFT for fixed w_0 summation = convolve result with “1”

10. Why is summation convolution?

11. start with definition

12. Why is summation convolution? Let

13. Why is summation convolution?

14. convolution with 1 equivalent to summation

15. DFT for fixed w_0 summation = convolve result with “1”

16. DFT for fixed w_0 output X(e^jw_0) is constant

17. DFT for fixed w_0

18. input signal x(n)

19. DFT for fixed w_0 fix specific frequency w_0

20. DFT for fixed w_0 modulate by complex exponential of frequency w_0

21. DFT for fixed w_0 summation = convolve with “1”

22. DFT for fixed w_0 Transfer function H(e^jw)

23. DFT for fixed w_0 summation = convolution with 1

24. DFT for fixed w_0 i.e. impulse response h(n) = 1 for all n

25. DFT for fixed w_0

26. output X(e^jw_0) is constant

27. Frequency Example

28. Arbitrary example

29. Frequency Example modulation = shift

30. Frequency Example convolution by 1 = multiplication by delta

31. DFT as filter bank

32. fix specific frequency w_0

33. DFT as filter bank one filter bank

34. DFT as filter bank w continuous between [0,2pi)

35. DFT as filter bank … uncountably many filter banks

36. DFT as filter bank … Uncountable  cannot enumerate all (even with infinite number of terms)

37. DFT as filter bank … bank of modulators of all frequencies between [0, 2pi)

38. DFT as filter bank … bank of identical filters with impulse response of h(n) = 1

39. Short-Time Fourier Transform

40. two variables

41. Short-Time Fourier Transform frequency

42. Short-Time Fourier Transform shift

43. Short-Time Fourier Transform shifted window function v(k)

44. Short-Time Fourier Transform let dummy variable be n instead of k

45. Short-Time Fourier Transform fix frequency w_0 and shift m

46. Short-Time Fourier Transform pass in input x(n)

47. Short-Time Fourier Transform multiply by shifted window and complex exponential

48. Short-Time Fourier Transform summation = convolve with 1

49. Short-Time Fourier Transform output constant determined by frequency w_0 and shift m

50. Short-Time Fourier Transform

51. fix frequency w_0 and shift m

52. Short-Time Fourier Transform pass in input x(n)

53. Short-Time Fourier Transform multiply by shifted window and complex exponential

54. Short-Time Fourier Transform summation = convolve with 1

55. Short-Time Fourier Transform output constant determined by frequency w_0 and shift m

56. Short-Time Fourier Transform

57. dummy variable k instead of n

58. Short-Time Fourier Transform shift is n (previously m)

59. Short-Time Fourier Transform rewrite

60. Short-Time Fourier Transform multiply by e^-jwn and e^jwn

61. Short-Time Fourier Transform n is shift variable

62. Short-Time Fourier Transform LTI system

63. Short-Time Fourier Transform Impulse response

64. Short-Time Fourier Transform flipped window modulated by +w

65. Short-Time Fourier Transform modulation by -w

66. Short-Time Fourier Transform

67. fixed shift n

68. Short-Time Fourier Transform fixed frequency w_0

69. Short-Time Fourier Transform convolution with modulated window Multiplication by shifted window transform freq domain

70. Short-Time Fourier Transform modulation by – w_0 shift by –w_0 freq domain

71. input X(e^jw)

72. transfer function (window transform shifted by +w_0)

73. LTI system output

74. final output after shift by –w_0

75. STFT as filter bank

76. fixed shift n

77. STFT as filter bank fixed frequency w_0

78. STFT as filter bank fixed shift n, fixed shift w_0 = one filter bank

79. STFT as filter bank

80. fixed shift n

81. STFT as filter bank let w vary between [0, 2pi)

82. STFT as filter bank uncountably many filters since w in [0, 2 pi)

83. STFT as filter bank bandpass filters separated by infinitely small shifts

84. transfer function (window transform shifted by +w_0)

85. STFT as filter bank demodulators

86. STFT as filter bank segments of X(e^jw)

87. final output after shift by –w_0

88. Fix number of frequencies

89. STFT as filter bank bank of M filters

90. STFT as filter bank M band pass filters

92. Uniformly spaced frequencies

93. STFT as filter bank bank of M filters becomes…

94. STFT as filter bank uniform DFT bank

95. M band pass filters …

96. STFT as filter bank Let E_k(z)=1 for all k

97. STFT as filter bank Let E_k(z)=1 for all k

98. STFT as filter bank window becomes rectangle M samples long ……

99. Uncertainty principle

100. wide window

101. Uncertainty principle narrow bandpass

102. Uncertainty principle wide window poor time resolution

103. Uncertainty principle narrow bandpass good frequency resolution

104. Uncertainty principle narrow window

105. Uncertainty principle wide bandpass

106. Uncertainty principle narrow window good time resolution

107. Uncertainty principle wide bandpass poor frequency resolution

108. Special case narrowest window

109. Special case widest bandpass

110. Special case narrowest window perfect time resolution

111. Special case widest bandpass no frequency resolution

112. v(n)=delta(n)

113. v(n)=delta(n)=1 if n=0, 0 otherwise

114. v(n)=delta(n)

115. STFT  DFT

116. v(m)=delta(m)

117. v(0)=delta(0)=1 STFT  DFT

118. LTI system output band limited

119. can decimate in time domain

120. Decimation in time domain

121. LTI system output band limited

122. Decimation in time domain maximal decimation (total of M samples across M channels)

123. Copies in frequency domain

124. LTI system output band limited

125. Copies in frequency domain copies after maximal decimation

126. Decimated STFT M M M

127. M M M uniformly spaced versions of same window filter

128. Decimated STFT M M M constant maximal decimation

129. Decimated STFT M M M decimation by M samples window shift of M

130. Decimated STFT (sliding window)

131. time axis

132. Decimated STFT (sliding window) frequency axis

133. Decimated STFT (sliding window) shift window by integer multiples of M

134. Decimated STFT M M M decimation by M samples window shift of M

135. Decimated STFT (sliding window) calculate M uniformly spaced samples of DFT

136. Decimated STFT M M M uniformly spaced versions of same window filter

137. Decimated STFT (sliding window)

138. time axis

139. Decimated STFT (sliding window) frequency axis

140. Decimated STFT (sliding window) shift window by integer multiples of M

141. Decimated STFT M M M decimation by M samples window shift of M

142. Decimated STFT (sliding window) calculate M uniformly spaces samples of DFT

143. Decimated STFT M M M uniformly spaced versions of same window filter

144. Decimated STFT (sliding window) uniform sampling of time/frequency

145. Decimated STFT (sliding window) M fixes sampling

146. Decimated STFT M M M decimation by M samples M versions of same window filter

147. Decimated STFT let decimation vary

148. Decimated STFT (sliding window) let window shifts vary

149. Decimated STFT let window transforms vary

150. Decimated STFT (sliding window) let window transforms vary

151. Wavelet Transform

153. let decimation vary

154. Wavelet Transform let window shifts vary

155. Wavelet Transform let window transforms vary

156. Wavelet Transform let window transforms vary

157. Wavelet Transform let window transforms vary

158. Summary

159. Decimated STFT M M M decimation by M samples M versions of same window filter

160. Decimated STFT (sliding window) M fixes sampling

161. Decimated STFT (sliding window) uniform sampling of time/frequency

162. Wavelet Transform let decimation vary let window transforms vary

163. Wavelet Transform let window shifts vary let window transforms vary

164. Wavelet Transform non-uniform sampling of time/frequency grid

165. Reference Multirate Systems and Filter Banks by P.P. Vaidyanthan, pp.457-486