Information about Digital Signal Processing Summary

3F3 – Digital Signal Processing (DSP), January 2009, lecture slides 8, Dr Elena Punskaya, Cambridge University Engineering Department

Introduction to DSP • Understand what is Digital Signal Processing • Be able to provide very briefly some examples of applications of DSP • Be able to state briefly main DSP limitations – aliasing (cannot distinguish between higher and lower frequencies, how to prevent – sampling theorem, correct reconstruction – antialias filter) – frequency resolution (sample for a limited period of time, does not pick up relatively slow changes) – quantisation error (sampling, loss of info, limited precision) • Be able to describe advantages of Digital over Analogue Signal Processing – reprogrammable / easily portable / duplicable – better control of accuracy – can be easily stored – precise mathematical operations 2

DTFT and DFT • Be aware of time-domain and frequency-domain analyses • Be comfortable with performing fundamental operations for sampled signals – DTFT, Inverse DTFT • Be able to state main problems with computing DTFT on a computer, explain how they can be overcome to obtain DFT • Be able to derive DFT from DFTF – by taking DFTF of the windowed signal • Be able to derive – spectrum of the windowed signal – rectangular window spectrum • Be aware of – zero-padding – Inverse DFT, circular convolution – Use of DFT and IDFT to compute standard convolution and thus perform linear filtering 3

FFT • Know the basic principles behind radix-2 FFT algorithms – N is a power of 2 – FFT butterfly structure – decomposition to reduce evaluation to single point DFT – bit reversal operations – in place computation – the number of computations required to compute one butterfly – the total number of stages required • Be able to show the total number of complex and real operation required to compute N-point FFT • Be able to demonstrate the efficiency of FFT compared to DFT (based on the total operations count) • Be able to five (briefly) examples of applications 4

Basics of Digital Filters • Be very familiar with the main characteristics – time-domain linear difference equations filter’s unit-sample (impulse) response (linear convolution causal LTI) – frequency-domain more general, Z-transform domain – system transfer function – poles and zeros diagram in the z-plane (stability) Fourier domain – frequency response (distance to poles and zeros, close to pole – magnitude rises, close to zero – magnitude falls) – spectrum of the signal • Be able to state and identify on the diagram main elements of Digital Filters – adders/multipliers/delays/advances • Be able to state four basic ideal filter types – lowpass/high-pass/band-pass/band-stop and their main characteristics – magnitude response and linear phase response • Be able to explain briefly why it is impossible to implement an ideal filter – needs to be causal to be realised 5

Design of FIR Filters • Know main characteristics – difference equation/transfer function/impulse response • Be aware of FIR using DFT and IDFT implementation • Know why linear phase filters are used/understand principles • Understand the window method for FIR filters – infinite response of the ideal filter and, hence, the need for truncation and shift to the right – truncation = pre-multiplication by rectangular window • a filter of large order has a narrow transition band • sharp discontinuity results in side-lobe interference – use of windows with no abrupt discontinuity can • Know how to use the window method for FIR filters (steps) • Be able to explain why the window method is not optimal – pass-band and stop-band parameters are equal thus unnecessary high accuracy in the pass band – the ripple of the window is not uniform – more freedom can be allowed Hence be able to give brief examples of other (optimal) methods of FIR filter design 6

Design of IIR filters • Know main characteristics – difference equation/transfer function/impulse response/stability issue • Be familiar with the main concepts of impulse invariant, matched z- transform and backward difference method and their disadvantages • Be able to state main properties of bilinear transform – produces a digital filter whose frequency response has the same characteristics as the frequency response of the analogue filter – maps the Left half s-plane onto the interior of the unit circle in the z-plane, ensures stability • monotonic Ω↔ ω mapping Ω= 0 is mapped to ω = 0, and Ω = ∞ is mapped to ω = π (half the sampling frequency). • • mapping between the frequency variables • Know how to use bilinear transform to design IIR filters (steps) • Know how to design highpass/bandpass/bandstop filters using frequency transformation • Be able to state the main problem with bilinear transform – performs a nonlinear mapping of the phase leading to a distortion (or warping) of the digital frequency response – hence pre-warping 7

Implementation of Digital Filters • Be able to compare IIR and FIR filters • Be able to state main concerns of filter implementation and ways of addressing them – Speed/power (+memory) • Be familiar with different forms of realization structures – Direct Form I/II – cascade/parallel/feedback and be able to briefly explain why they are of use • Be able to state the undesirable consequences of finite-precision filter implementation and explain the strategies for overcoming them – Overflow (scaling and saturation arithmetic) • Be familiar with roundoff (quantisation) noise generation, limit cycles and deadbands 8

Thank you! 9

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