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What you should know after these lectures?

Elena Punskayawww-sigproc.eng.cam.ac.uk/~op205

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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 – antialiasfilter)

– 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

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• 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

DTFT and DFT

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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

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Basics of Digital Filters

• Be very familiar with the main characteristics– time-domain

linear difference equationsfilter’s unit-sample (impulse) response (linear convolution causal LTI)

– frequency-domainmore 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

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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 allowedHence be able to give brief examples of other (optimal) methods of FIR filter design

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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

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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/feedbackand 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

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Thank you!