ALARI/DSP INTRODUCTION -2

1. ALARI/DSPINTRODUCTION-2 Toon van Waterschoot & Marc Moonen Dept. E.E./ESAT, K.U.Leuven toon.vanwaterschoot@esat.kuleuven.be http://homes.esat.kuleuven.be/~tvanwate

2. INTRODUCTION-1 : Overview • Introduction • Discrete-time signals sampling, quantization, reconstruction • Stochastic signal theory deterministic & random signals, (auto-)correlation functions, power spectra, … • Discrete-time systems LTI, impulse response, FIR/IIR, causality & stability, convolution & filtering, … • Complex number theory complex numbers, complex plane, complex sinusoids, circular motion, sinusoidal motion, … Toon van Waterschoot & Marc Moonen INTRODUCTION-2

3. INTRODUCTION-2 : Overview • z-transform and Fourier transform region of convergence, causality & stability, properties, frequency spectrum, transfer function, pole-zero representation, … • Elementary digital filters shelving filters, presence filters, all-pass filters • Discrete transforms DFT, FFT, properties, fast convolution, overlap-add/overlap-save, … Toon van Waterschoot & Marc Moonen INTRODUCTION-2

4. z- and Fourier-transform: overview • z-transform: • definition & properties • complex variables • region of convergence • Fourier transform: • frequency response • Fourier transform • Transfer functions: • difference equations • rational transfer functions • poles & zeros • stability in the z-domain Toon van Waterschoot & Marc Moonen INTRODUCTION-2

5. discrete-time sequence in integer variable z-transform discrete-time series in complex variable z- and Fourier-transform: z-transform • definition: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

6. z-transform z- and Fourier-transform: z-transform • definition: • z-transform of a discrete-time signal: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

7. z-transform z- and Fourier-transform: z-transform • definition: • z-transform of a discrete-time system impulse response: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

8. z- and Fourier-transform: z-transform • properties: • linearity property: • time-shift theorem: • convolution theorem: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

9. z- and Fourier-transform: z-transform • region of convergence: • the z-transform of an infinitely long sequence is a series with an infinite number of terms • for some values of the series may not converge • the z-transform is only defined within the region of convergence (ROC): Toon van Waterschoot & Marc Moonen INTRODUCTION-2

10. z- and Fourier-transform: Fourier transf. • Frequency response: • for an LTI system a sinusoidal input signal produces a sinusoidal output signal at the same frequency • the output can be calculated from the convolution: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

11. z- and Fourier-transform: Fourier transf. • Frequency response: • the sinusoidal I/O relation is • the system’s frequency response is a complex function of the radial frequency : • denotes the magnitude response • denotes the phase response Toon van Waterschoot & Marc Moonen INTRODUCTION-2

12. z- and Fourier-transform: Fourier transf. • Frequency response: • the frequency response is equal to the z-transform of the system’s impulse response, evaluated at • for , is a complex function describing the unit circle in the z-plane Im z-plane Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

13. z- and Fourier-transform: Fourier transf. • Frequency response & Fourier transform • the frequency response of an LTI system is equal to the Fourier transform of the continuous-time impulse sequence constructed with h[k] : • similarly, the frequency spectrum of a discrete-time signal (=its z-transform evaluated at the unit circle) is equal to the Fourier transform of the continuous-time impulse sequence constructed with u[k], y[k] : • Input/output relation: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

14. z- and Fourier-transform: Transfer func. • Difference equations: • the I/O behaviour of an LTI system using an FIR model, can be described by a difference equation: • the I/O behaviour of an LTI system using an IIR model, can be described by a difference equation with an autoregressive part in the left-hand side: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

15. z- and Fourier-transform: Transfer func. • Rational transfer functions: • transforming the FIR difference equation to the z-domain and using the convolution theorem, leads to: • the z-transform of the impulse response is called the transfer function of the system: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

16. z- and Fourier-transform: Transfer func. • Rational transfer functions: • transforming the IIR difference equation to the z-domain and using the convolution theorem, leads to: • the ratio of and is equal to the z-transform of the impulse response and is called the transfer function of the system: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

17. Im z-plane Re z- and Fourier-transform: Transfer func. • Poles and zeros: • the zeros of a rational transfer function are defined as the roots of the nominator polynomial • the poles of a rational transfer function are defined as the roots of the denominator polynomial • e.g. Toon van Waterschoot & Marc Moonen INTRODUCTION-2

18. Im unstable Re z- and Fourier-transform: Transfer func. • Stability in the z-domain: • the pole-zero representation of a rational transfer function allows for an easy stability check • an LTI system is stable if all of its poles lie inside the unit circle in the complex z-plane Im stable Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

19. INTRODUCTION-2 : Overview • z-transform and Fourier transform region of convergence, causality & stability, properties, frequency spectrum, transfer function, pole-zero representation, … • Elementary digital filters shelving filters, presence filters, all-pass filters • Discrete transforms DFT, FFT, properties, fast convolution, overlap-add/overlap-save, … Toon van Waterschoot & Marc Moonen INTRODUCTION-2

20. Elementary digital filters: overview • Shelving filters: • definition • one-zero • one-pole • Presence filters: • definition • two-zero • two-pole • biquadratic • All-pass filters: • definition • biquadratic Toon van Waterschoot & Marc Moonen INTRODUCTION-2

21. Elementary digital filters: shelving filters • Definition: • a shelving filter is a filter that amplifies a signal in the frequency range Hz (boost), while attenuating it in the range Hz (cut), or vice versa • Low-pass filter: • low-frequency boost, high-frequency cut • High-pass filter: • low-frequency cut, high-frequency boost • Cut-off frequency: • the cut-off frequency is usually defined as the frequency at which the filter gain is 3dB less than the gain at Hz (low-pass) or Hz (high-pass) Toon van Waterschoot & Marc Moonen INTRODUCTION-2

22. Elementary digital filters: shelving filters • One-zero shelving filter: • difference equation: • transfer function: • signal flow graph:  Toon van Waterschoot & Marc Moonen INTRODUCTION-2

23. Elementary digital filters: shelving filters • One-zero shelving filter: • 1 real zero: • highpass if • lowpass if Im Im highpass lowpass Re Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

24. Elementary digital filters: shelving filters • One-zero shelving filter: • frequency response • frequency magnitude response: • frequency phase response: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

25. Elementary digital filters: shelving filters • One-zero shelving filter: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

26. Elementary digital filters: shelving filters • One-pole shelving filter: • difference equation: • transfer function: • signal flow graph:  Toon van Waterschoot & Marc Moonen INTRODUCTION-2

27. Elementary digital filters: shelving filters • One-pole shelving filter: • 1 real pole: • highpass if • lowpass if Im Im highpass lowpass Re Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

28. Elementary digital filters: shelving filters • One-pole shelving filter: • frequency response • frequency magnitude response: • frequency phase response: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

29. Elementary digital filters: shelving filters • One-pole shelving filter: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

30. Elementary digital filters: presence filters • Definition: • a presence filter is a filter that amplifies a signal in the frequency range around a center frequency Hz (boost), while attenuating elsewhere (cut), or vice versa • Resonance filter: • boost at center frequency (band-pass) • Notch filter: • cut at center frequency (band-stop) • Bandwidth: • the bandwidth is defined as the frequency difference between the frequencies at which the filter gain is 3dB lower/higher than the resonance/notch gain Toon van Waterschoot & Marc Moonen INTRODUCTION-2

31. Elementary digital filters: presence filters • Two-zero presence filter: • diff. eq.: • transfer function: • signal flow graph:   Toon van Waterschoot & Marc Moonen INTRODUCTION-2

32. Elementary digital filters: presence filters • Two-zero presence filter: • 2 zeros: • if : real zeros  cascade shelving filters • if : complex conj. zero pair  notch filter Im Im cascade shelving filters notch filter Re Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

33. Elementary digital filters: presence filters • Two-zero notch filter: • transfer function in radial representation: • radial center frequency • zero radius Im notch filter Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

34. Elementary digital filters: presence filters • Two-zero notch filter: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

35. Elementary digital filters: presence filters • Two-pole presence filter: • diff. eq.: • transfer function: • signal flow graph:   Toon van Waterschoot & Marc Moonen INTRODUCTION-2

36. Elementary digital filters: presence filters • Two-pole presence filter: • 2 poles: • if : real poles  cascade shelving filters • if : comp. conj. pole pair  resonance filter Im Im cascade shelving filters resonance filter Re Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

37. Elementary digital filters: presence filters • Two-pole resonance filter: • transfer function in radial representation: • radial center frequency • pole radius Im resonance filter Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

38. Elementary digital filters: presence filters • Two-pole resonance filter: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

39. Elementary digital filters: presence filters • Biquadratic presence filter: • difference equation: • transfer function: • 2 poles: • 2 zeros: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

40. Elementary digital filters: presence filters • Biquadratic presence filter: • signal flow graph:   Toon van Waterschoot & Marc Moonen INTRODUCTION-2

41. Elementary digital filters: presence filters • Constrained biquadratic presence filter: constrained biquadratic resonance filter constrained biquadratic notch filter Im Im Re Re Toon van Waterschoot & Marc Moonen INTRODUCTION-2

42. Elementary digital filters: all-pass filters • Definition: • a (unity-gain) all-pass filter is a filter that passes all input signal frequencies without gain or attenuation • hence a (unity-gain) all-pass filter preserves signal energy • an all-pass filter may have any phase response Toon van Waterschoot & Marc Moonen INTRODUCTION-2

43. Elementary digital filters: all-pass filters • Biquadratic all-pass filter: • it can be shown that for the unity-gain constraint to hold, the denominator coefficients must equal the numerator coefficients in reverse order, e.g., • the poles and zeros are moreover related as follows Toon van Waterschoot & Marc Moonen INTRODUCTION-2

44. INTRODUCTION-2 : Overview • z-transform and Fourier transform region of convergence, causality & stability, properties, frequency spectrum, transfer function, pole-zero representation, … • Elementary digital filters shelving filters, presence filters, all-pass filters • Discrete transforms DFT, FFT, properties, fast convolution, overlap-add/overlap-save, … Toon van Waterschoot & Marc Moonen INTRODUCTION-2

45. Discrete transforms: overview • Discrete Fourier Transform (DFT): • definition • inverse DFT • matrix form • properties • Fast Fourier Transform (FFT): • Digital filtering using the DFT/FFT: • linear & circular convolution • overlap-add method • overlap-save method • fast convolution Toon van Waterschoot & Marc Moonen INTRODUCTION-2

46. Discrete transforms: DFT • DFT definition: • the Fourier transform of a signal or system is a continuous function of the radial frequency : • the Fourier transform can be discretized by sampling it at discrete frequencies , uniformly spaced between and : = DFT Toon van Waterschoot & Marc Moonen INTRODUCTION-2

47. Discrete transforms: DFT • Inverse discrete Fourier transform (IDFT): • an -point DFT can be calculated from an -point time sequence: • vice versa, an -point time sequence can be calculated from an -point DFT: = IDFT Toon van Waterschoot & Marc Moonen INTRODUCTION-2

48. Discrete transforms: DFT • matrix form • using the shorthand notations the DFT and IDFT definitions can be rewritten as: DFT: IDFT: Toon van Waterschoot & Marc Moonen INTRODUCTION-2

49. Discrete transforms: DFT • matrix form • the DFT coefficients can then be calculated as • an -point DFT requires complex multiplications Toon van Waterschoot & Marc Moonen INTRODUCTION-2

50. Discrete transforms: DFT • matrix form • the IDFT coefficients can then be calculated as • an -point IDFT requires complex multiplications Toon van Waterschoot & Marc Moonen INTRODUCTION-2