slides!

Home page of EDISP
(English) DIgital Signal Processing course
winter 2009/10

Schedule

There are lab exercises, 4 hours every second week, room 022 (basement).

Labs will be on Mondays, 8-12 , in two subgroups of not more than 12 students.
For the introductory lab (lab0) we met on 12 Oct, 9:15 room 022. You will be able to register for subgroups then.
Next lab (lab1) will be

• 19.10.2009 (8:15-12) for N subgroup,
• 26.10.2009 (8:15-12) for P subgroup

After the Independence Day (11.11) "N" subgroup will have labs on weeks marked as "N" in the official elka calendar (and "P" on "P" weeks). First three labs are different....

Books

Book base

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing, Prentice-Hall 1989 (or II ed, 1999; also acceptable previous editions entitled Digital Signal Processing).

Other books

A free textbook covering some of the subjects can be found here: http://www.dspguide.com/pdfbook.htm
The book is slightly superficial, but it can be valuable - at least as a quick reference.

• Edmund Lai, Practical Digital Signal Processing for Engineers and Technicians, Newnes (Elsevier), 2003
  seems also a simple but thoroughly written book.

Additional books available in Poland:

• R.G. Lyons, Wprowadzenie do cyfrowego przetwarzania sygnałów (WKiŁ 1999)
• Craig Marven, Gilian Ewers, Zarys cyfrowego przetwarzania sygnałów, WKiŁ 1999 (simple, slightly too easy) [en: A simple approach to digital signal processing, Wiley & Sons, 1996]
• Tomasz P. Zieliński, Od teorii do cyfrowego przetwarzania sygnałów, WKiŁ 2002 (and next edition with slightly modified title)

Please remember:

• there are notation differences between lecture and "dspguide"
• The official book is Oppenheim & Schafer (though notation is sometimes different too)
• no book is obligatory as it is hard to get O&S, and other books do not cover the subject fully.

Probably the best choice is to buy a used copy of O&S. It'll serve you for years, if you are interested in DSP. And it contains a lot of PROBLEMS to solve and learn!

Or you may prefer to buy/borrow a laboratory scriptbook for CYPS, which is in Polish language (Cyfrowe Przetwarzanie Sygnałów, red. A Wojtkiewicz, Wydawnictwa PW).
 

Lecture slides

• Lecture 1 slides: newlect1.pdf but please don't read the schedule there - use the schedule.pdf
• Lecture 2 slides: newlect2.pdf
  Convolution example: conv_exampl.jpg
• Lecture 3 slides: newlect3.pdf
• Lecture 4 slides newlect4.pdf
  Windowing for DFT, DFT applications slides: newlect5.pdf
  
• Lecture 5:
  FFT from newlect4.pdf.
  Convolution by FFT from newlect5.pdf
  Instantaneous spectrum: newlect6.pdf

  Homework will be given this day!

• Lecture 6:(10.11)
  Lecture "Linear difference equations" - last 3 slides from newlect2.pdf
  and Z-transform newlect7.pdf
  Beginning of "Filters part I" newlect8.pdf
  Homework (hand-written on paper, worth up to 2 points) is due at noon on Friday, November 13th (room 453). You may submit it earlier (preferably at the lecture).
  
• Lecture 7(17.11): Review before test I - based on homework errors.
  
• Lecture 8(24.11): newlect7.pdf and
  Filters part I:newlect8.pdf
  
• Lecture 9(1.12): (++1) hour test I, 10pts worth: bring YOUR OWN notes (handwritten on paper or on printed lecture slides). No books, no photocopies of other person notes. Example test:test1_078a.pdf
  
• Lecture 10(8.12):
  last slides of Filters part I
  Filters part II:newlect9.pdf
  Filters part IIInewlect10.pdf (and maybe more...)
• Lecture 11(15.12): Digital signal processors: lect12_dsp.pdf(and OHP foils to be seen at the lecture)
  Homework:(homew2_2008plus.pdf) given today
• 22.12: no lecture ("Thursday")
• 29.12: Merry Christmas and Happy New Year
• Lecture 12 (5.1): 2D signal processing lect13.pdf
  Homework due today
• Lecture 13 (12.1): test II( example test here) and signal processing for data compression
• Lecture 14: (19.1) advanced techniques including signal processing for data compression
• Lecture 15: (26.1)Exam 0: pen, pencil, calculator and your own notes plus lecture slides.
  Copies of solutions for homeworks/tests/exams are NOT allowed.

  The exam covers ALL the course matter. There are "Problems" (longer) and "Questions" (shorter), for total of 90 minutes.
  Exam 0 is a "bonus" - if you fail you are still entitled to take Ex1 and Ex2 and no "bad" record remains.
  Students who earned the "shortpath" grade may take the ex0 or ex1 without any risk - better grade counts

Examples of final tests (historical)

One test. Another test.

There is no guarantee that the current test be identical ;-). It will be similar (the lecture was similar), but I might also put more focus on different subjects. The only base is the lecture content (live one, not only the published slides ....).

The main rule: exam covers the whole course content (sampled), including the T1(H1)+T2(H2) area and also the lectures after the H2.

Lab info: example lab exercises

Disclaimer:

Openly speaking, they are exercise sets current at the time of posting. I reserve the right to make some important modifications before the actual lab, to give different sets to different groups etc. (and I usually DO review the text before giving it....).

• Lab 0: Introduction, Matlab
• Lab 1: Signals, systems, frequency
  • Simple MATLAB usage: make a vector, plot x - y plot with proper data on axes, make a simple m-function.
  • DT signal as a sampled CT signal: plot sample values of a sin() with a frequency of 1 kHz, sampled with 10 kHz (etc). Put x-axis values as sample index, CT instant, ....
  • Normalized frequency concept (e.g. What is the θ value in the above example?)
  Lab exercises: nlab1.pdf
• Lab 2: Spectral analysis (+ continuation of lab1)
  • Impulse response of a system; initial conditions etc.
  • DFT properties, effect of limited observation time (windowing)
  • Spectrum of a rectangular impulse
  Lab exercises: nlab2.pdf
• Lab 3: Instantaneous spectrum Lab exercises: lab03.pdf
  Example entry test:
  • Calculate the Fourier spectrum of a rectangular window with given length. Find the number of \{zero places|sidelobes\} and mainlobe width .
  • Calculate the DFT of a signal consisting of exactly 4 periods of a cosine, or 2 periods zero-padded to the length of 4 periods.
  • Calculate the STFT (inst. spectrum) of a u(t)cos(0.2π n) for a rectangular window of given length positioned at n=0, n=9, ....
• Lab 4: Filter design Lab exercises: lab4.pdf
  Example entry test:
  • Location of two poles and two zeros is given. Calculate filter frequency characteristics. (Hints: a) start from easy configurations, e.g. zeros on the unit circle, double pole at z=0; b) First recall which root configurations are possible for a real-coefficient polynomial, then recall stability criterion for DT filter)
  • A filter structure is given. Find all its properties: h(n), H(z), poles/zeros, amplitude characteristics..... (for own exercises - use all the lecture examples first)
  • A filter is characterized by H(z) (given). Find the output y(n) for x(n)=sin(θ₁n)+cos(θ₂n)+1 (or some other combination of different signals)
  • Explain step-by-step design process of FIR filter by windowing the ideal filter characteristics. Explain role of window (etc.).
• Lab 5: we don't do it (I'm just using old numbering scheme with 7 labs....)
• Lab 6: Signal processors Lab exercises: lab6.pdf

Homework solutions

Test2 solutions

Exam 0 solutions

Past things archive (Attic)

1. In both cases the signal was sampled correctly (f_s>2f)
   1. To calculate N₀ it was enough to count no. of samples in period (or divide f_s over f). Answer was 10(A) or 6(B). For N₀ samples in period, θ₀ was equal to 2π/N₀.
   2. K-size DFT will have K discrete samples over <-π, +π) (we include -π, and exclude +π , but due to periodicity of spectrum it is only a convention)
      for a cosine, only two samples are non-zero: at k such that θ_k=±θ_signal. Form the definition of θ_k you will see that this is for k=±4 (this is the result of taking K=4N₀).
   3. You may label frequency axis with k=-K/2,....-1,0,1,...K/2 or with its periodic equivalent K-K/2,....K-1,0,1,...K/2
      to label with θ just use the expression for θ_k.
2. 1. H(z)=Y(z)/X(z) is easily obtainable from the time equation. It was
      0.2(1+z^-1)/(1-0.8z^-1) [A]
      -0.2(1-z^-1)/(1+0.8z^-1) [B]
   2. Zeros are roots of numerator: -1 [A] or +1 [B]
      Poles are roots of denominator: +0.8 [A] or -0.8 [B]. They are inside unit circle, so the system is stable (but I didn't ask...)
   3. Example graph: - please find a_0, b_0, b_1 by yourselves. If you are smart, you may save one multiplication by 0.2 (this is left as exercise to you).
   4. For x(n)= shifted delta, (a limited energy signal) you may take the impulse response and shift it appropriately. To find h(n) it is easiest to split H(z) into two fractions: (shown for [A], for [B] change some signs)
      0.2(1/(1-0.8z^-1)+z^-1/(1-0.8z^-1))
      and lookup the inv.Z of 1/(1-0.8z^-1) in the table. The final result is a sum of two identical exponentials shifted by 1 in time. Then, you shift h(n) to proper position....
   5. For x(n) = 1-(-1)ⁿ (a periodic signal) we see a DC component and a periodic component exp(jπn) with frequency of π. We find numerical values of H(0)=(2 or 0) and H(π)= (0 or -2) by substituting exp(0) and exp(π) for z, and finally
      y(n)=H(0)-H(π)·(-1)ⁿ
3. The response was symmetrical around its midpoint (n= 2 or 4). Thus, it was a repsonse of a zero-phase filter delayed by 2 or 4.
   1. phase is linear φ=-(2 or 4)θ
      delay is constant and equal to (2 or 4)
   2. The response of filter is a rectangle modulated by exp(jπn). Thus, the characteristics is a sin(θL/2)/sin(θ), shifted to π. You may find the mainlobe width, you may plot exactly zeros of A(θ) etc.
4. Time resolution is proportional to time duration of window, frequency resolution - to mainlobe width (which is prop. to 1/K ....).
   Rectangular window has narrowest mainlobe possible, but high sidelobes; so it is good for resolving signals close in frequency, but without large difference in amplitudes.
   Any other window will have wider mainlobe (so poorer resolution in f).
5. 1. There was nonlinearity introcuced by product of two samples (linear is multiplication by a constant only).
      Saying causal=yes because of BIBO was not enough; because FIR was enough; if you call BIBO, you have to prove it by finding relation between bound of input and output.
   2. LP filter with passband of π/4 (see the "lecture 17").
   3. Many shorter is better: by averaging we reduce the variance of estimate. (variance is huuuuuge with single FFT)
   4. β (some call it α) controls the shape of window - effectively the sidelobes level (high β - low sidelobes).
   5. Inv FT is calculated by summation when the spectrum is discrete ([B], periodic signal) and by integration when the spectrum is continuous ([A], limited energy signal).
   6. 3 buses are for opcode, data1 (signal), data2 (coefficient).
      Any instruction with dual move uses all three, e.g MAC instruction needs 2 data, so it is nice that we can load data in the same cycle in 56002 it can be coded as:
      mac a,b x:(r0+),x0 y:(r4+),y0
   7. Trivial
   8. def: order of n^2, FFT: order of n log2(n)
   9. y(n) length is, maximally, (length of h(n))+(length of x(n))-1. K=M+N-1. Here, we were asked to find M knowing K and N. Answer is, as you may guess, M=K-(N-1)
   10. The clue is in word "maximally". It may happen that for certain signal (e.g in the stopband....) the y(n) is shorter.....