Advanced Digital Signal Processing

Lecture "Advanced Digital Signal Processing"

Basic Information
Lecturers:	Gerhard Schmidt (lecture) and Christian Kanarski (exercise)
Room:	Building F, room F-SR-II
E-mail:	This email address is being protected from spambots. You need JavaScript enabled to view it.
Language:	English
Target group:	Students in electrical engineering and computer engineering
Prerequisites:	Basic Knowledge about signals and systems
Contents:	Students attending this lecture should be able to implement efficient and robust signal processing structures. Knowledge about moving from the analog to the digital domain and vice versa including the involved effects (and trap doors) should be acquired. Also differences (advantages and disadvantages) between time and frequency domain approaches should be learnt. Topic overview: Digital processing of continuous-time signals Efficient FIR structures DFT and FFT Digital filters (FIR filters/IIR filters) Multirate digital signal processing
References:	J. G. Proakis, D. G. Manolakis: Digital Signal Processing: Principles, Algorithms, and Applications, Prentice Hall, S. K. Mitra: Digital Signal Processing: A Computer-Based Approach, McGraw Hill Higher Education, 2000 A. V. Oppenheim, R. W. Schafer: Discrete-time signal processing, Prentice Hall, 1999, 2nd edition M. H. Hayes: Statistical Signal Processing and Modeling, John Wiley and Sons, 1996

News

There will be a question/answer session on Friday the 07.02. starting from 09:15 at the usual lecture/exercise room (Building F, room F-SR-II). Send in your questions regarding topics of the lecture/exercise to This email address is being protected from spambots. You need JavaScript enabled to view it.. Only questions that were sent in beforehand will be answered in full detail.

Lecture Slides

Link	Content
	Slides of the lecture "Introduction" (Contents, literature, analog versus digital signal processing)
	Slides of the lecture "Digital processing of continuous-time signals" (Sampling, Quantization, AD and DA conversion)
	Slides of the lecture "Efficient FIR structures" (DFT and signal processing, FFT, DFT of real sequences)
	Slides of the lecture "DFT/FFT" (DFT and signal processing, FFT, DFT of real sequences)
	Slides of the lecture "Digital filters" (FIR filters, IIR filters, analysis and design)
	Slides of the lecture "Multi-rate digital signal processing" (Upsampling, downsampling, filter banks)

Exercises

Please note that the questionnaires will be uploaded every week before the excercises, if you download them earlier, you won't get the most recent version.

Link	Content	Solution
	Exercise 1: Sampling
	Exercise 2: Quantization
	Exercise 3: Discrete Fourier Transform (DFT) / Convolution
	Exercise 4: FFT / Radix-2-FFT
	Exercise 5: FFT / FFTs of Real and Complex Signals
	Exercise 6: Signal Flow Graph
	Exercise 7: Round-off Effects in Digital Filters
	Exercise 8: FIR Filter Design
	Exercise 9: IIR Filter Design
	Exercise 10: Multirate Digital Signal Processing

Evaluation

The completed evaluations of the last years can be found here. Please note that this page contains the evaluations of all of our lectures. As a consequence you must scroll down to get to the evaluations of this lecture.

Matlab demos

Matlab file for the comparison of window sequences (click to expand)

%**************************************************************************
% Comparison of "standard" and a bit more "advanced" window functions
%**************************************************************************

%**************************************************************************
% Basis parameters
%**************************************************************************
N_win =   64;
N_dft = 1024*16;

%**************************************************************************
% Basic windows - rectangle window
%**************************************************************************
h_rec     = ones(N_win,1);
h_rec     = h_rec / sum(h_rec);

H_rec     = fft(h_rec,N_dft);
H_rec     = H_rec(1:N_dft/2+1);
H_rec_log = 20*log10(abs(H_rec)+eps);

%**************************************************************************
% Basic windows - Hann window
%**************************************************************************
h_han     = hann(N_win);
h_han     = h_han / sum(h_han);

H_han     = fft(h_han,N_dft);
H_han     = H_han(1:N_dft/2+1);
H_han_log = 20*log10(abs(H_han)+eps);

%**************************************************************************
% Basic windows - Hamming window
%**************************************************************************
h_ham     = hamming(N_win);
h_ham     = h_ham / sum(h_ham);

H_ham     = fft(h_ham,N_dft);
H_ham     = H_ham(1:N_dft/2+1);
H_ham_log = 20*log10(abs(H_ham)+eps);

%**************************************************************************
% Advanced windows - "Chebyshev" window
%**************************************************************************
h_che     = chebwin(N_win,10);
h_che     = h_che / sum(h_che);

H_che     = fft(h_che,N_dft);
H_che     = H_che(1:N_dft/2+1);
H_che_log = 20*log10(abs(H_che)+eps);

%**************************************************************************
% Advanced windows - "Prolate" window
%**************************************************************************
h_pro     = dpss(N_win,1.18);
h_pro     = h_pro / sum(h_pro);

H_pro     = fft(h_pro,N_dft);
H_pro     = H_pro(1:N_dft/2+1);
H_pro_log = 20*log10(abs(H_pro)+eps);

%**************************************************************************
% Show results
%**************************************************************************
fig = figure(1);
f = (0:N_dft/2)/N_dft*2;

plot(f,H_rec_log,'b', ...
     f,H_han_log,'r', ...
     f,H_ham_log,'k', ...
     f,H_che_log,'m', ...
     f,H_pro_log,'c', ...
     'LineWidth',2);
legend('Rectangle window', ...
       'Hann window', ...
       'Hamming window', ...
       'Chebyshev window', ...
       'Prolate spheroidal window')
grid on;
xlabel('Normalized frequency \(\Omega/\pi\)','interpreter','latex');
ylabel('dB')
ylim([-90 10])

Matlab file for the effects of quantization on filter design (click to expand)

%**************************************************************************
% Design parameters
%**************************************************************************
N   =  8;     % Filter order
f_c =  0.1;   % Normalized cut-off frequency (0 ... 1)
R_p =  0.5;   % Ripple in dB in passband
R_s = 80;     % Stopband attenuation in dB

%**************************************************************************
% Design of an elliptic lowpass filter
%**************************************************************************
[b,a] = ellip(N, R_p, R_s, f_c);

%**************************************************************************
% Show frequency response
%**************************************************************************
fig = figure(1);
set(fig,'Units','Normalized');
set(fig,'Position',[0.1 0.1 0.8 0.8]);
[H,Omega] = freqz(b,a,2048*4,'whole',2);
plot(Omega,20*log10(abs(H)+eps),'b','LineWidth',2);
grid on
axis([0 2 (-R_s -20) 20])
xlabel('Normalized frequency \Omega/\pi')
ylabel('dB')


%**************************************************************************
% Quantization
%**************************************************************************
B = 32; % Number of bits

a_max = max(abs(a));
b_max = max(abs(b));

a_q = round(a / a_max * 2^B) / 2^B * a_max;
b_q = round(b / b_max * 2^B) / 2^B * b_max;

%**************************************************************************
% Show frequency response of quantized filter
%**************************************************************************
[H_q,Omega] = freqz(b_q,a_q,2048*4,'whole',2);
hold on;
plot(Omega,20*log10(abs(H_q)+eps),'r','LineWidth',2);
hold off;
legend('Non-quantized',['Quantized with ',num2str(B),' bits'])

%**************************************************************************
% Show coefficients
%**************************************************************************
format long;
a
a_q
b
b_q

%**************************************************************************
% Transform to cascade of biquad filters
%**************************************************************************
[sos,g] = tf2sos(b,a);

[L,L_tmp] = size(sos);

sos_q = round(sos / max(max(abs(sos))) * 2^B) / 2^B * max(max(abs(sos)));
g_q   = round(g^(1/L) * 2^B) / 2^B;

H_bq_q = freqz(g_q*sos_q(1,1:3),sos_q(1,4:6),2048*4,'whole',2);
for k = 2:L
    H_bq_q = H_bq_q .* freqz(g_q*sos_q(k,1:3),sos_q(k,4:6),2048*4,'whole',2);
end;

%**************************************************************************
% Show frequency response of quantized biquad filters
%**************************************************************************
[H_q,Omega] = freqz(b_q,a_q,2048*4,'whole',2);
hold on;
plot(Omega,20*log10(abs(H_bq_q)+eps),'k','LineWidth',2);
hold off;
legend('Non-quantized',['Quantized with ',num2str(B),' bits'],...
       'Biquad structure (also qunatized)');

C Examples

C file with different realizations of an FIR filter (click to expand)

#include <math.h>
#include <stdlib.h>
#include <time.h>
#include <stdio.h>

// Parameters (as defines)
#define FILTERLENGTH 1000
#define SIGNALLENGTH 1000000

// Parameters (as variables)
int iRingbufferOffset;

// Vector declarations
float afInput[SIGNALLENGTH];
float afSigBuffer[FILTERLENGTH];
float afCoeffBuffer[FILTERLENGTH];

// Output vector declarations
float afOutput0[SIGNALLENGTH];
float afOutput1[SIGNALLENGTH];
float afOutput2[SIGNALLENGTH];
float afOutput3[SIGNALLENGTH];
float afOutput4[SIGNALLENGTH];

float fLocResult;
float fTmp;

// Time declarations
clock_t tStart, tEnd;
double dTime;

// Counter
int n, m, k;

// Float random number from -1.0 to 1.0
float float_rand()
{
   float max = 1.0;
   float min = -1.0;
   float scale = (float)rand() / (float)RAND_MAX; /* [0, 1.0] */
   return min + scale * (max - min);       /* [min, max] */
}

// Main function
int main()
{
   // Initialize arrays
   for (m = 0; m < FILTERLENGTH; m++)
   {
      afSigBuffer[m] = 0.0;
      afCoeffBuffer[m] = (float)1.0 / (float)(m + 1);
   }
   for (n = 0; n < SIGNALLENGTH; n++)
   {
      afInput[n] = float_rand();
      afOutput0[n] = 0.0;
      afOutput1[n] = 0.0;
      afOutput2[n] = 0.0;
      afOutput3[n] = 0.0;
      afOutput4[n] = 0.0;
   }

   /*-------------------------------------------------------------------------------*
    * Linear buffer with direct memory write
    *-------------------------------------------------------------------------------*/

   // Start timer
   tStart = clock();

   // Clear input buffer
   for (m = 0; m < FILTERLENGTH; m++)
   {
      afSigBuffer[m] = 0.0;
   }

   for (n = 0; n < SIGNALLENGTH; n++)
   {
      // Update linear buffer
      for (m = FILTERLENGTH - 1; m > 0; m--)
      {
         afSigBuffer[m] = afSigBuffer[m - 1];
      }
      afSigBuffer[0] = afInput[n];

      // Calculate filter output
      for (m = 0; m < FILTERLENGTH; m++)
      {
         afOutput0[n] += afSigBuffer[m] * afCoeffBuffer[m];
      }
   }

   // End timer and print time difference
   tEnd = clock();
   dTime = ((double)tEnd - (double)tStart) / (double)CLOCKS_PER_SEC;
   printf("Time for linear buffer with direct memory write is:   %f s\n", dTime);

   /*-------------------------------------------------------------------------------*
    * Linear buffer with local result variable
    *-------------------------------------------------------------------------------*/

   // Start timer
   tStart = clock();

   // Clear input buffer
   for (m = 0; m < FILTERLENGTH; m++)
   {
      afSigBuffer[m] = 0.0;
   }

   for (n = 0; n < SIGNALLENGTH; n++)
   {
      // Update linear buffer
      for (m = FILTERLENGTH - 1; m > 0; m--)
      {
         afSigBuffer[m] = afSigBuffer[m - 1];
      }
      afSigBuffer[0] = afInput[n];

      fLocResult = 0.0;
      // Calculate filter output
      for (m = 0; m < FILTERLENGTH; m++)
      {
         fLocResult += afSigBuffer[m] * afCoeffBuffer[m];
      }
      afOutput1[n] = fLocResult;
   }

   // End timer and print time difference
   tEnd = clock();
   dTime = ((double)tEnd - (double)tStart) / (double)CLOCKS_PER_SEC;
   printf("Time for linear buffer with local result variable is: %f s\n", dTime);

   /*-------------------------------------------------------------------------------*
    * Ringbuffer with if-condition
    *-------------------------------------------------------------------------------*/

   // Start timer
   tStart = clock();

   // Initialize ringbuffer offset
   iRingbufferOffset = FILTERLENGTH - 1;

   // Clear input buffer
   for (m = 0; m < FILTERLENGTH; m++)
   {
      afSigBuffer[m] = 0.0;
   }

   for (n = 0; n < SIGNALLENGTH; n++)
   {
      // Update buffer
      afSigBuffer[iRingbufferOffset] = afInput[n];

      // Calculate filter output
      fLocResult = 0.0;
      for (m = 0, k = iRingbufferOffset; m < FILTERLENGTH; m++)
      {
         fLocResult += afCoeffBuffer[m] * afSigBuffer[k];

         if (++k >= FILTERLENGTH)
         {
            k = 0;
         }
      }
      afOutput2[n] = fLocResult;

      // Update iRingbufferOffset
      if (--iRingbufferOffset < 0)
      {
         iRingbufferOffset = FILTERLENGTH - 1;
      }
   }

   // End timer and print time difference
   tEnd = clock();
   dTime = ((double)tEnd - (double)tStart) / (double)CLOCKS_PER_SEC;
   printf("Time for ringbuffer is:                               %f s\n", dTime);

   /*-------------------------------------------------------------------------------*
    * Ringbuffer with partitioned convolution
    *-------------------------------------------------------------------------------*/

   iRingbufferOffset = FILTERLENGTH - 1;

   // Start timer
   tStart = clock();

   // Clear input buffer
   for (m = 0; m < FILTERLENGTH; m++)
   {
      afSigBuffer[m] = 0.0;
   }

   for (n = 0; n < SIGNALLENGTH; n++)
   {
      // Update ringbuffer
      afSigBuffer[iRingbufferOffset] = afInput[n];

      fLocResult = 0.0;
      // Calculate first part of filter output (0 ... iRingbufferOffset)
      for (m = 0, k = iRingbufferOffset; k < FILTERLENGTH; m++, k++)
      {
         fLocResult += afSigBuffer[k] * afCoeffBuffer[m];
      }

      // Calculate second part of filter output
      for (k = 0; m < FILTERLENGTH; m++, k++)
      {
         fLocResult += afSigBuffer[k] * afCoeffBuffer[m];
      }

      afOutput3[n] = fLocResult;

      // Update iRingbufferOffset
      if (--iRingbufferOffset < 0)
      {
         iRingbufferOffset = FILTERLENGTH - 1;
      }
   }

   // End timer and print time difference
   tEnd = clock();
   dTime = ((double)tEnd - (double)tStart) / (double)CLOCKS_PER_SEC;
   printf("Time for ringbuffer with partitioned convolution is:  %f s\n", dTime);

   /*-------------------------------------------------------------------------------*
    * Ringbuffer with unrolling by hand
    *-------------------------------------------------------------------------------*/

   iRingbufferOffset = FILTERLENGTH - 1;

   // Start timer
   tStart = clock();

   // Clear input buffer
   for (m = 0; m < FILTERLENGTH; m++)
   {
      afSigBuffer[m] = 0.0;
   }

   for (n = 0; n < SIGNALLENGTH; n++)
   {
      // Update ringbuffer
      afSigBuffer[iRingbufferOffset] = afInput[n];

      fLocResult = 0.0;
      // Calculate first part of filter output (0 ... iRingbufferOffset)
      k = iRingbufferOffset;
      m = 0;
      for (; k < FILTERLENGTH - 6; m += 6, k += 6)
      {
         fLocResult += (afSigBuffer[k] * afCoeffBuffer[m] + afSigBuffer[k + 1] * afCoeffBuffer[m + 1] + afSigBuffer[k + 2] * afCoeffBuffer[m + 2] + afSigBuffer[k + 3] * afCoeffBuffer[m + 3] + afSigBuffer[k + 4] * afCoeffBuffer[m + 4] + afSigBuffer[k + 5] * afCoeffBuffer[m + 5]);
      }
      for (; k < FILTERLENGTH; k++, m++)
      {
         fLocResult += afSigBuffer[k] * afCoeffBuffer[m];
      }

      // Calculate second part of filter output
      k = 0;
      for (; m < FILTERLENGTH - 6; m += 6, k += 6)
      {
         fLocResult += (afSigBuffer[k] * afCoeffBuffer[m] + afSigBuffer[k + 1] * afCoeffBuffer[m + 1] + afSigBuffer[k + 2] * afCoeffBuffer[m + 2] + afSigBuffer[k + 3] * afCoeffBuffer[m + 3] + afSigBuffer[k + 4] * afCoeffBuffer[m + 4] + afSigBuffer[k + 5] * afCoeffBuffer[m + 5]);
      }
      for (; m < FILTERLENGTH; k++, m++)
      {
         fLocResult += afSigBuffer[k] * afCoeffBuffer[m];
      }

      afOutput4[n] = fLocResult;

      // Update iRingbufferOffset
      if (--iRingbufferOffset < 0)
      {
         iRingbufferOffset = FILTERLENGTH - 1;
      }
   }

   // End timer and print time difference
   tEnd = clock();
   dTime = ((double)tEnd - (double)tStart) / (double)CLOCKS_PER_SEC;
   printf("Time for ringbuffer unrolling by hand is:             %f s\n", dTime);

   /*-------------------------------------------------------------------------------*
    * Check for errors in convolution (linear buffer as assumed true result)
    *-------------------------------------------------------------------------------*/

   float fConvTest1 = (float)0.0;
   float fConvTest2 = (float)0.0;
   float fConvTest3 = (float)0.0;
   float fConvTest4 = (float)0.0;

   for (n = 0; n < SIGNALLENGTH; n++)
   {
      fConvTest1 += (float)fabs(afOutput0[n] - afOutput1[n]);
      fConvTest2 += (float)fabs(afOutput0[n] - afOutput2[n]);
      fConvTest3 += (float)fabs(afOutput0[n] - afOutput3[n]);
      fConvTest4 += (float)fabs(afOutput0[n] - afOutput4[n]);
   }
   printf("\n");
   printf("Summed absolute error for ringbuffer with direct memory writing is: %f\n", fConvTest1);
   printf("Summed absolute error for ringbuffer is: %f\n", fConvTest2);
   printf("Summed absolute error for ringbuffer with partitioned convolution is: %f\n", fConvTest3);
   printf("Summed absolute error for unrolling by hand is: %f\n", fConvTest4);

   return 0;
}

Exams

Remember to register for the exam in the QiS system! And book a time-slot for your oral exam using this form.

Visit of the Hans Böckler Foundation

The Hans Böckler Foundation offers students not only financial support, but also a wide range of seminars. We had the pleasure of taking part in an exciting seminar on the topic of ‘Data channels in the seabed - insights into the underwater infrastructure of the future’. We spent one day at the Faculty of Engineering at Kiel University. We were given presentations on various topics. Such as the geology of the seabed and the submarine cable incident between Finland and Estonia.

We were particularly impressed by the opportunity to take a look behind the scenes and experience the work of the students and researchers up close. We were allowed to visit the clean room and the special ‘paddling pool’ where experiments are tested directly in water. We were deeply impressed by the university and its diverse research opportunities. Thank you very much for your hospitality and the exciting insights!

Text and photo by Luise Artmann