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

1. What you should know after these
lectures?
Elena Punskaya
www-sigproc.eng.cam.ac.uk/~op205
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2. 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
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3. 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
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4. 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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5. 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
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6. 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

7. 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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8. 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
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9. Thank you!
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