SCHOOL of ENGINEERING
MSc in Signal Processing and Communications
Digital Signal Processing Labs (PGEE11108)
Marked Assignment 1 Digital sampling and aliasing
Submission Deadline: 4pm 21st October 2019
ALL SUBMISSIONS MUST BE THE STUDENTS OWN INDIVIDUAL WORK. SUBMISSIONS WILL BE CHECKED USING ANTI-PLAGARISM SOFTWARE. PLAGARISM AND COPYING WILL BE SUBJECT TO PENALTY.
Instructions
This assignment forms the first part of the marked assessment process for Digital Signal Processing Labs course PGEE11108. The completed solution for this assignment must be submitted via Learn by the deadline above. All solution reports should be in the form of a word document. Submissions should be made up of two elements:
1. A short report of no more than 2 pages (excluding appendix) describing your understanding of the problem and provide a detailed description of your proposed solutions and observations. Any figures included should be fully and correctly labelled and within the 2 pages
2. An appendix that contains the MATLAB code created in the exercise which should be fully commented
Marks will be awarded for the following components of the work: Overall presentation 10%, Problem understanding 50%, and Results and Conclusions 40%.
NOTE: MATLAB has specific routines for interpolation and resampling signals and resizing images. These are NOT to be used for the purposes of this assignment. However, the conv and conv2 functions may be used.
Digital sampling and aliasing
Background
Digital signals and images are often provided with standard sampling rates (audio) or pixel sizes/resolutions (images). For example, high quality music is usually sampled at 44 kHz while for telephone quality speech it is sufficient to sample between 5-8 kHz. Low resolution computer images are often digitized to VGA resolution (640 x 480) while digital camera images for an 8 Megapixel camera are 3264 x 2448.
Moving from one sampling rate to another in audio signals or changing the resolution of an image is called digital resampling and a very useful tool in DSP. This lab will investigate the task of resampling, including up-sampling, down-sampling and anti-aliasing. During this assignment you will see that the effects of aliasing and that the challenges of digital resampling are very similar to those of analogue sampling and reconstruction.
Changing the sampling rate
When the sampling frequencies are integer valued then a common strategy for re-sampling is to:
1. Up sampling: interleave the original samples with a suitable number of zeros.
2. Anti-aliasing filter: filter the signal to remove any frequencies that may cause aliasing, and then
3. Down sampling: reduce the sampling down to the target rate by only retaining the appropriate samples.
PGEE111081
Assignment
Up sampling
1) Generate the following signal of duration 1 second with a fundamental frequency of 𝑓 = 200Hz and 9 harmonics: 10
𝑥(𝑡)= ∑cos(2𝜋𝑘𝑓𝑡), k=1
sampled at 5 kHz (you can listen to the signal using the soundsc command). Confirm that it has the frequency spectrum that you expect. Without using MATLAB’s built-in resampling functions design a function to re-sample this signal to 8 kHz sampling frequency. Your solution will require you to make various design choices along the way – please use your initiative and explain your choices. Demonstrate and explain in your report the working of your function and any design considerations that you have made.
Resampling and filtering
2) Modify your code from part (1) to generate a filtered version of the original signal using the impulse response function of
an intercom which is given in the file telephoneIR.wav (in the resources directory), which is sampled at 22 kHz. The output signal should be sampled at 8 kHz as in part (1). Your report should explain all the steps involved and demonstrate the success of your method.
Fractional Delay
3) Another important resampling operation in digital sampling (e.g. in audio processing) is to delay signals by a fraction of a sample period. Briefly discuss in your report how you might design a filter to do this using sampling theory.
PGEE111082