1. ALARI/DSP INTRODUCTION-2
Toon van Waterschoot & Marc Moonen
Dept. E.E./ESAT, K.U.Leuven

2. INTRODUCTION-1 : Overview
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. z- and Fourier-transform: z-transform
definition:
discrete-time sequence in integer variable
z-transform
discrete-time series in complex variable
Toon van Waterschoot & Marc Moonen INTRODUCTION-2

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

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

18. 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 Im Re
unstable
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:
two-zero
two-pole
biquadratic
All-pass filters:
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:
Elementary digital filters: presence filters
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

51. Discrete transforms: DFT
properties:
linearity & time-shift theorem (cf. z-transform)
frequency-shift theorem (modulation theorem):
circular convolution theorem:
if and are periodic with period , then
(see also ‘Digital filtering using the DFT/FFT’)
Toon van Waterschoot & Marc Moonen INTRODUCTION-2

52. Discrete transforms: FFT
Fast Fourier Transform (FFT)
divide-and-conquer approach:
split up N-point DFT in two N/2-point DFT’s
split up two N/2-point DFT’s in four N/4-point DFT’s …
split up N/2 2-point DFT’s in N 1-point DFT’s
calculate N 1-point DFT’s
rebuild N/2 2-point DFT’s from N 1-point DFT’s
rebuild two N/2-point DFT’s from four N/4-point DFT’s
rebuild N-point DFT from two N/2-point DFT’s
DFT complexity of multiplications is reduced to FFT complexity of multiplications
James W. Cooley
Carl Friedrich Gauss (
John W.Tukey
Toon van Waterschoot & Marc Moonen INTRODUCTION-2

53. Discrete transforms: Digital filtering
Linear and circular convolution:
circular convolution theorem: due to the sampling of the frequency axis, the IDFT of the product of two -point DFT’s corresponds to the circular convolution of two length- periodic signals
LTI system: the output sequence is the linear convolution of the impulse response with the input signal
Toon van Waterschoot & Marc Moonen INTRODUCTION-2

54. Discrete transforms: Digital filtering
Linear and circular convolution:
the linear convolution of a length impulse response with a length- input signal is equivalent to their -point circular convolution if both sequences are zero-padded to length :
zero padding
Toon van Waterschoot & Marc Moonen INTRODUCTION-2

55. Discrete transforms: Digital filtering
Overlap-add/overlap-save:
in many applications the input sequence length is much larger than the impulse response length
computing the DFT of a very long or even infinitely long sequence is not feasible
a block-based convolution method is more appropriate:
the input sequence is divided in relatively short blocks
each input block is circularly convolved with the impulse response using the DFT approach
the output signal is reconstructed using the overlap-add method or the overlap-save method
Toon van Waterschoot & Marc Moonen INTRODUCTION-2

56. Discrete transforms: Digital filtering
Overlap-add method: … + +
+ …
Toon van Waterschoot & Marc Moonen INTRODUCTION-2

57. Discrete transforms: Digital filtering
Overlap-save method: = = …
discard
discard …
discard
Toon van Waterschoot & Marc Moonen INTRODUCTION-2

58. Discrete transforms: Digital filtering
Fast convolution:
if in the above digital filtering methods the DFT is implemented using an FFT algorithm, then so-called fast convolution methods are obtained
Toon van Waterschoot & Marc Moonen INTRODUCTION-2

59. Hungry for more? Some nice introductory books:
S. J. Orfanidis, “Introduction to Signal Processing”, Prentice-Hall Signal Processing Series, 798 p., 1996
J. H. McClellan, R. W. Schafer, and M. A. Yoder, “DSP First: A Multimedia Approach”, Prentice-Hall, 1998
P. S. R. Diniz, E. A. B. da Silva and S. L. Netto, “Digital Signal Processing: System Analysis and Design”, Cambridge University Press, 612 p., 2002
Some interesting online books:
Smith, J.O. Mathematics of the Discrete Fourier Transform (DFT), , ISBN
Smith, J.O. Introduction to Digital Filters, August 2006 Edition,
Toon van Waterschoot & Marc Moonen INTRODUCTION-2