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20-6554: Digital Signal Processing

20-6554: Digital Signal Processing. SUMMARY. General remarks. 1. Go through the exercise sheets (the questions at the end of each chapter) together with the sample solutions. 2. Review the PPT slides. 3. Go through the assignment questions and model solutions.

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20-6554: Digital Signal Processing

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  1. 20-6554: Digital Signal Processing SUMMARY

  2. General remarks 1. Go through the exercise sheets (the questions at the end of each chapter) together with the sample solutions. 2. Review the PPT slides 3. Go through the assignment questions and model solutions 4. Look at the past exam questions and solutions 5. Use the following ‘highlights’ as a guide to the main parts of each chapter 6. Use of supplied software (programs and spreadsheets) as necessary

  3. Chapter 1: Discrete Time Signals Definitions Recursive, non-recursive Shannon's Sampling Theorem "An analog signal containing components up to some maximum frequency f1 Hz may be completely represented by regularly spaced samples, provided the sampling rate is at least 2f1 samples per second” Frequency in terms of Hz and radians per second Various parts of waveforms, and their mathematical descriptions: amplitude, period, phase.

  4. Chapter 1: Discrete Time Signals Jargon Nyquist frequency Maximum frequency f1 contained in the analog signal. Nyquist rate Minimum sampling rate (2f1 samples per second) at which the signal can be recovered Folding frequency Half the sampling frequency, the highest frequency which can be represented. The spectrum of a sampled signal repeats around multiples of the sampling frequency. If the signal contains frequencies higher than the folding frequency, then repetitions of the spectrum overlap and cause distortion when you try to reconstruct the signal. This is known as ALIASING.

  5. Chapter 1: Discrete Time Signals Basic Types of Digital Signal Unit step: Unit impulse: Unit ramp: Note that and and that and

  6. Chapter 1: Discrete Time Signals Periodic signals A second problem concerns the frequency scale. As indicated above, there are 2/ samples per period (dimensionless frequency scale). If we do need to use a frequency scale, we can use the actual sampling interval T to give the sampling frequency: The digital signal is then .

  7. Chapter 1: Discrete Time Signals Linear system: obeys the principle of superposition - if input leads to output (aiconstants) then input leads to output A consequence is the property of frequency preservation (the output can only contain those frequencies present in the input) Time invariant means that system characteristics do not change with time i.e. Shifting the input shifts output by the same amount.

  8. Chapter 1: Discrete Time Signals Other system properties Terms worth knowing about: Causality Output signal depends only on present and/or previous values of input. Stability Produces a finite or bounded response to a bounded input. Invertibility If a system turns input x[n] into output y[n], then its inverse (if it exists) turns input y[n] into output x[n]. Memory A system with memory can calculate its output at step n using not only x[n] but also previous values x[n-1], x[n-2] etc.

  9. Describing digital signals with impulse functions Chapter 2: Time -Domain Analysis Impulse-response, step-response Convolution – calculation by flip/shift and using the formula Concept of a difference equation

  10. Definition of discrete Fourier series for periodic time series Chapter 3: Frequency -Domain Analysis Calculation of the spectra by evaluating each term, and by using complex exponentials Sketch the amplitude (magnitude) and phase

  11. Chapter 3: Frequency -Domain Analysis Definition and calculation of the Fourier Transform of aperiodic digital sequences Determination of the frequency response H(Ω) as the ratio of Y(Ω) to X(Ω). These three functions are the FT of h[n], y[n] and x[n] respectively. Determination of H(Ω) using the formula: Sketch H(Ω)

  12. Chapter 4: The z-transform Definition z as a time-shift operator Convolution in time-domain equivalent to multiplication in frequency-domain Derivation of x[n] using recursive algorithm, using Z-plane poles and zeros, and the interpretation in terms of the frequency response – the geometrical interpretation Stability of filter represented by a particular expression Minimum-delay First and second-order systems

  13. Chapter 5: Design of non-recursive digital filters Definition of non-recursive The Fourier-transform method Creation of high/band pass filter from low-pass prototype Windowing – triangular, von Hann, Hamming windows

  14. Chapter 6: Design of recursive digital filters Definition of recursive Designs based on z-plane poles and zeros Specification of bandwidth leading to pole radius Filters derived from analog designs – Butterworth, Chebyshev Determination of filter order from cutoff frequency requirements

  15. Chapter 7: The Discrete and Fast Fourier Transforms The DFT, and its similarity to the discrete Fourier series Frequency-sampling theorem, and its implications - the number of points in a spectrum Implementation of the DFT in Excel, and the effect of real input data Relationship between real/imaginary, even/odd input data, and the nature of the resulting spectrum: For real x[n]: even x[n] => real and even spectrum odd x[n] => imaginary and odd spectrum For imaginary x[n]: even x[n] => imaginary and even spectrum odd x[n] => real and odd spectrum next slide …

  16. Chapter 8: FFT Processing The FFT – what it is and how it is used. Using a programmed tool and using the Excel built-in function For real x[n], spectrum is symmetric, so the number of points in the spectrum = N/2. Determination of df from dt and N. Exact harmonics; causes and meaning of spectral spreading Explanation of spectral leakage, and the calculation of the relative amplitudes of spectral peaks Spectrum of a signal in noise; effect on spectrum when the data are not ‘mean-zero’ next slide …

  17. Chapter 8 (contd): FFT Processing Windowing, and its use in improving the spectrum Tapered windows De-trending – how (and why) it is done Zero-filling (NOT a smug statement about dental hygiene) – why and how it is done Digital filtering by Fast Convolution next slide …

  18. Chapter 9: Random Digital Signals Concepts of deterministic and non-deterministic Reasons for modelling signals with random processes - difficulty of analogue approach - need randomness to convey information - noise is random; require methods to enhance signals in noise Measure basic statistical properties of signals: mean, mean-square, variance Ensemble and time-averages Effect of estimating these from a time-limited data set next slide …

  19. Chapter 9 (contd): Random Digital Signals Autocorrelation – its definition and use. Autocovariance, Symmetry. ACF of random data, finite and infinite sequence Power spectrum, as FFT of ACF - including pre-processing (windowing etc) Cross-correlation. Asymmetry. Use in determining time differences between similar signals Use in identifying common shared frequencies Cross-spectrum next slide …

  20. Chapter 10: Random DSP How statistical measures of signals are affected by processing Calculation of these after processing Optimal designs for enhancing a signal in noise Results: e.g - the mean of the output is the mean of the input multiplied by the sum of the impulse response terms. - other results are summarised in the PPT for chapter 10 Calculations of these – e.g. problems 10.1, 10.2, 10.3, 10.4 Results of passing white noise through a filter System identification by cross-correlation next slide …

  21. Chapter 10 (continued) : Random DSP Three techniques for the three objectives of the chapter: 1. Signal recovery from noise. Can predict (quantify) reduction in noise power after filtering 2. Signal detection. Can determine whether a specific signal is present, and where it occurs, by using a matched-filter. 3. Enhancement of a repetitive signal by averaging over many repetitions. Pseudo-Random Binary Sequence (PSBR) System identification by cross-correlation

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