Lecture 5. The Digital Fourier Transform. (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith)

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1 Lecture 5 The Digital Fourier Transform (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith)

2 Why bother? The ear processes sound by decomposing it into sine waves at different frequencies. A machine that does the same would be a step towards one that hears things as humans do. So how do we do this by machine? Peripheral Auditory system 11 Hz Cochlea, Auditory nerve 66 Hz 22 Hz

3 Jean Baptiste Joseph Fourier A French mathematician and physicist who lived from presented a paper in 187 to the Institut de France claiming any continuous periodic signal could be represented as the sum of properly chosen sinusoidal waves. Among the reviewers were two famous mathematicians Joseph Louis Lagrange ( ) Pierre Simon de Laplace ( ) Lagrange said sine waves could not perfectly represent signals with discontinuous slopes, like square waves. (He was, technically, right) Thus, the Institut de France did not publish Fourier s work until 15 years later, after Lagrange died.

4 The Fourier Series Given a periodic function x(t). xt ( + mt) = xt ( ) m the period A series of sine and cosine functions reproduces x(t) = + + n n n n n= 1 [ ] xt A A ω t B ω t () cos( ) sin( ) Where the frequency ω n of each sine and cosine is an integer multiple (n) of a fundamental frequency ω o ω ω π = n = 2 n/ T n

5 Fundamental Frequency The lowest frequency that a sine or cosine can have and still fit exactly into one period of the function. f = ω /2 π = 1/ T

6 Fourier Series Quite a few! The frequency of this component xt () A cos( ω t) = A+ n n B sin( ω t) n= 1 + n n The original signal Amplitude: the contribution of this component

7 Kinds of Fourier Transforms Fourier Transform Signals are continuous and aperiodic Fourier Series Signals are continuous and periodic Discrete Time Fourier Transform Signals are discrete and aperiodic What we use! Discrete Fourier Transform Signals are discrete and periodic

8 Digital Sampling An analog signal is sampled into sequence of discrete sample points, x[n] AMPLITUDE sample interval quantization increment TIME

9 Window function A function that is zero-valued outside of some chosen interval. When a signal (data) is multiplied by a window function, the product is zero-valued outside the interval: all that is left is the "view" through the window.

10 Some famous windows Rectangular wn= [ ] 1 amplitude sample Hann (Julius von Hann) Note: we assume w[n] = outside some range [,N] [ ] wn Bartlett [ ] wn 2π n =.5 1 cos N 1 2 N 1 N 1 = i n N amplitude amplitude sample sample

11 Windowing x[n] is windowed by function w[n] (multiply the ith value of x by the ith value of w) x[n] w[n] z[n] x = Example: windowing x[n] with a rectangular window

12 Why window shape matters Don t forget that a DFT assumes the signal in the window is periodic The boundary conditions mess things up unless you manage to have a window whose length is exactly 1 period of your signal Making the edges of the window less prominent helps suppress undesirable artifacts

13 Only what s in the window Do the DFT only on the values in the window z[n] x[n] w[n] Ignore Ignore x =

14 Discrete Fourier Transform Represents a finite sequence of complex values as a finite number of discrete real and imaginary sinusoids Time domain xn [ ] Real portion N-1 Imaginary portion N-1 DFT IDFT Frequency domain X[ n] Real portion N/2 N-1 Imaginary portion N/2 N-1

15 Discrete Fourier Transform A series of complex amplitudes in the time domain become a series of complex amplitudes in the frequency domain Time domain xn [ ] Real portion N-1 Imaginary portion N-1 DFT IDFT Frequency domain Xk [ ] Real portion N/2 N-1 Imaginary portion N/2 N-1

16 Discrete Fourier Transform If the time-domain signal has no imaginary part (like an audio signal) then the frequency-domain signal is symmetric around N/2..so we can (mostly) ignore frequencies over N/2 Time domain xn [ ] Real portion N-1 Imaginary portion N-1 DFT IDFT Frequency domain Xk [ ] Real portion N/2 N-1 Imaginary portion N/2 N-1

17 Some numbers The highest frequency represented depends on the sample rate of the signal Number of points in the sample window More sample points = finer frequency resolution Sample rate n s + 1 frequencies spaced evenly from to 2 2

18 2 Fundamental Frequencies Fundamental frequency of analysis : based number of points in the window & the sample rate Fundamental frequency of the signal : based on the period of the signal. n s + 1 frequencies spaced evenly from to 2 2

19 Questions If n = 16 and s = 8 Hz, what is the fundamental frequency of analysis? what frequencies are represented in the DFT? What if n = 32 and s = 8 Hz? What if n = 32 and s = 16 Hz? n s + 1 frequencies spaced evenly from to 2 2

20 Discrete Fourier Transform A 16 point real-valued sequence is represented by (yes we re ignoring the ones above N/2) (16/2)+1 = 9 Cosine Waves and (16/2)+1 = 9 Sine Waves

21 Complex Numbers z x+ iy A y ( cosθ isin θ ) A + x x = Acos θ y = Asin θ ( 2 2) A= x + y

22 Euler s Formula Useful for relating polar coordinates to rectangular coordinates e iθ = cosθ + i sin θ Thus... z = Ae iθ PHASE AMPLITUDE

23 Multiplying Complex Numbers POLAR notation EASIER for this ( ) i θ z = Ae z = Ae z = AA e i ( θ + θ ) 1 2 i θ

24 Multiplying Complex Numbers Cartesian works as follows z = x + iy z = x + iy ( ) z = xx yy + i xy + x y

25 Complex Conjugate the complex conjugate of a complex number is given by changing the sign of the imaginary part. A complex number Its complex conjugate z = a+ ib z = a ib

26 Back to the DFT Discrete Fourier Transform Inverse Discrete Fourier Transform What s different between them N 1 Xk [ ] = xne [ ] n= N 1 k = 2πi kn N 1 xn [ ] = X[ ke ] N 2πi N kn

27 Put in Cartesian Coordinates N 1 N 1 Xk [ ] = xne [ ] n= 2πi kn N 2πkn 2πkn = xn [ ] cos + isin N N n= REMEMBER EULER S FORMULA REMEMBER COMPLEX MULTIPLICATION

28 You can code this up! N 1 2πkn 2πkn Xk [ ] = xn [ ] cos + isin n= N N Time domain xn [ ] Real portion N-1 Imaginary portion N-1 DFT IDFT Frequency domain Xk [ ] Real portion N/2 N-1 Imaginary portion N/2 N-1

29 The Inverse DFT N 1 N 1 1 xn [ ] = X[ ke ] N complex number k = 2πi kn N 1 2πkn 2πkn = Xk [ ] cos + isin N N N k = Seriously, remember complex multiplication here. You ll need it.

30 You can code this up! N 1 1 2πkn 2πkn xn [ ] = X[ k] cos + isin N k = N N Time domain xn [ ] Real portion N-1 Imaginary portion N-1 DFT IDFT Frequency domain Xk [ ] Real portion N/2 N-1 Imaginary portion N/2 N-1

31 What about N/2? I said that you can (mostly) ignore frequencies over N/2. What is up with that? Let s have a look. When the input signal in the time domain x[n] is all real values, the signal in the frequency domain X[n] is symmetric around N/2

32 Computational complexity How many operations does this take for each frequency? How many operations total? N 1 2πkn 2πkn Xk [ ] = xn [ ] cos + isin N N n=

33 The FFT Fast Fourier Transform A much, much faster way to do the DFT Introduced by Carl F. Gauss in 185 Rediscovered by J.W. Cooley and John Turkey in 1965 The Cooley-Turkey algorithm is the one we use today (mostly) Big O notation for this is O(N log N) Matlab functions fft and ifft are standard.

34 Short time Fourier Transform Break signal into windows Calculate DFT of each window

35 The Spectrogram spectrogram(y,256,128,256,fs,'yaxis'); A series of short term DFTs Typically just displays the magnitudes in X of the frequencies up to ½ sampe rate There is a spectrogram function in matlab

36 The Spectrogram spectrogram(y,124,512,124,fs,'yaxis'); A series of short term DFTs Typically just displays the magnitudes in X of the frequencies up to ½ sampe rate There is a spectrogram function in matlab

Chapter 4 Discrete Fourier Transform (DFT) And Signal Spectrum

Chapter 4 Discrete Fourier Transform (DFT) And Signal Spectrum CEN352, DR. Nassim Ammour, King Saud University 1 Fourier Transform History Born 21 March 1768 ( Auxerre ). Died 16 May 1830 ( Paris ) French