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Brief Review of Fourier Analysis


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3F3 – Digital Signal Processing (DSP), January 2009, lecture slides 2, Dr Elena Punskaya, Cambridge University Engineering Department

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Brief Review of Fourier Analysis

  1. 1. Brief Review of Fourier Analysis Elena Punskaya Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner 22
  2. 2. Time domain Example: speech recognition difficult to differentiate between different sounds in time domain tiny segment sound /i/ as in see sound /a/ as in father 23
  3. 3. How do we hear? Cochlea – spiral of tissue with liquid and thousands of tiny hairs that Inner Ear gradually get smaller Each hair is connected to the nerve The longer hair resonate with lower frequencies, the shorter hair resonate with higher frequencies Thus the time-domain air pressure signal is transformed into frequency spectrum, which is then processed by the brain Our ear is a Natural Fourier Transform Analyser! 24
  4. 4. Fourier’s Discovery Jean Baptiste Fourier showed that any signal could be made up by adding together a series of pure tones (sine wave) of appropriate amplitude and phase (Recall from 1A Maths) Fourier Series for periodic square wave infinitely large number of sine waves is required 25
  5. 5. Fourier Transform The Fourier transform is an equation to calculate the frequency, amplitude and phase of each sine wave needed to make up any given signal : (recall from 1B Signal and Data Analysis) 26
  6. 6. Prism Analogy Analogy: a prism which splits white light into a spectrum of colors White Spectrum light of colours white light consists of all frequencies mixed together Signal Fourier Spectrum the prism breaks them apart so we can see the Transform separate frequencies 27
  7. 7. Signal Spectrum Every signal has a frequency spectrum. • the signal defines the spectrum • the spectrum defines the signal We can move back and forth between the time domain and the frequency domain without losing information 28
  8. 8. Time domain / Frequency domain •  Some signals are easier to visualise in the frequency domain •  Some signals are easier to visualise in the time domain •  Some signals are easier to define in the time domain (amount of information needed) •  Some signals are easier to define in the frequency domain (amount of information needed) Fourier Transform is most useful tool for DSP 29
  9. 9. Fourier Transforms Examples signal spectrum cosine t ω added higher frequency t ω component Back to our sound recognition problem: peaks correspond to t the resonances of sound /a/ ω the vocal tract shape as in father in logarithmis units of dB they can be used to sound /i/ t differentiate between as in see ω sounds in logarithmis units of dB 30
  10. 10. Discrete Time Fourier Transform (DTFT) What about sampled signal? The DTFT is defined as the Fourier transform of the sampled signal. Define the sampled signal in the usual way: Take Fourier transform directly using the “sifting property of the δ-function to reach the last line 31
  11. 11. Discrete Time Fourier Transform – Signal Samples Note that this expression known as DTFT is a periodic function of the frequency usually written as The signal sample values may be expressed in terms of DTFT by noting that the equation above has the form of Fourier series (as a function of ω) and hence the sampled signal can be obtained directly as [You can show this for yourself by first noting that (*) is a complex Fourier series with coefficients however it is also covered in one of Part IB Examples Papers] 32
  12. 12. Computing DTFT on Digital Computer The DTFT expresses the spectrum of a sampled signal in terms of the signal samples but is not computable on a digital computer for two reasons: 1.  The frequency variable ω is continuous. 2.  The summation involves an infinite number of samples. 33
  13. 13. Overcoming problems with computing DTFT The problems with computing DTFT on a digital computer can be overcome by: Step 1. Evaluating the DTFT at a finite collection of discrete frequencies. no undesirable consequences, any frequency of interest can always be included in the collection Step 2. Performing the summation over a finite number of data points does have consequences since signals are generally not of finite duration 34
  14. 14. The Discrete Fourier Transform (DFT) The discrete set of frequencies chosen is arbitrary. However, since the DTFT is periodic we generally choose a uniformly spaced grid of N frequencies covering the range ωT from 0 to 2π. If the summation is then truncated to just N data points we get the DFT The inverse DFT can be used to obtain the sampled signal values from the DFT: multiply each side by and sum over p=0 to N-1 Orthogonality property of complex exponentials is N if n=q and 0 otherwise 35
  15. 15. The Discrete Fourier Transform Pair 36
  16. 16. Properties of the Discrete Fourier Transform (DFT) •  is periodic, for each p •  is periodic, for each n •  for real data [You should check that you can show these results from first principles] 37
  17. 17. DTFT – Normalised Frequency Please also note the DTFT and IDTFT pair is often written as: The assumption here is that ω is a normalized frequency ω=2πfΤ = 2π(f/fs) - normalized frequency (rad/sample) f - cycles per second fs - samples per second f/fs - cycles per sample 38 We will adopt this notation for majority of the slides.