Many problems in digital signal processing can be solved much more easily in the frequency domain than in the time domain. Therefore we begin with a repetition of the Fourier transform. In order to process time continuous signals in a digital computer they have to be sampled. This leads us to the well known sampling theorem and the reconstruction of analog signals from samples using low pass filtering. The discrete and fast Fourier transform are used to implement digital filters efficiently. Parallel to the lectures a system is developed to transmit two signals over a common channel in different frequency bands using modulation.
You can download the following Java programs and start them with double click or java -jar program.jar from the command line. All you need is a free Java SDK installation.
- Convolution convolution.jar
- Real Fourier series, fundamental and harmonic frequencies fourierreihe.jar
- Fourier series, rotating complex phasors zeiger.jar
- Fourier series of a complex function, rotating complex phasors fourierreihekomplex.jar
- Transition from Fourier series to Fourier transform fourierlimit.jar
- Fourier transform of a finite length signal fourierintervall.jar
- Sampling theorem abtastthm.jar
- Sampling and aliasing for a cosine wave samplingalias.jar
- Signal reconstruction lowpass.jar
- Lowpass filter in image processing bild.jar
- Discrete Fourier Transform dft.jar
- Spectrogram, modulation spektrogramm.jar
- Vocoder, modulation vocoder.jar
- Cyclic and linear convolution zyklisch.jar
- Fast convolution with FFT schnellefaltung.jar
- Hilbert Transform hilbert.jar
I will upload homework every week here. Some of the exercises are milestones for the project. Try to do them regularly in order to avoid too much workload at the deadline of the project.
Part of the lecture is a project, which has to be accomplished in a team with maximum 3 persons. Project presentations are scheduled in January. Deadline for submission of a written report (8 pages) is December 31. Please attach the check list.
Fourier transform, sampling, aliasing, signal reconstruction
- Lecture notes Mathematics 2 (ASE)
- J. H. McClellan, R. W. Schafer, M. A. Yoder: Signal Processing First, Pearson 2003
- Single side band modulation, Hilbert transform
FFT, fast convolution
- A. V. Oppenheim, R. W. Schafer: Discrete-Time Signal Processing, Prentice Hall 1989
- B. Meffert, O. Hochmuth: Werkzeuge der Signalverarbeitung, Pearson 2004
- E. O. Brigham: FFT-Anwendungen, Oldenbourg 1997
- Video: invention of the FFT