Signal processing Signals may have to be transformed in order to Digital Signal Processing → amplify or filter out embedded information → detect patterns Markus Kuhn → prepare the signal to survive a transmission channel → prevent interference with other signals sharing a medium → undo distortions contributed by a transmission channel → compensate for sensor deficiencies → find information encoded in a different domain Computer Laboratory To do so, we also need http://www.cl.cam.ac.uk/teaching/1213/DSP/ → methods to measure, characterise, model and simulate trans- mission channels → mathematical tools that split common channels and transfor- Michaelmas 2012 – Part II mations into easily manipulated building blocks 3 Signals Analog electronics → flow of information Passive networks (resistors, capacitors, R inductances, crystals, SAW filters), → measured quantity that varies with time (or position) non-linear elements (diodes, . . . ), U in L C U out (roughly) linear operational amplifiers → electrical signal received from a transducer (microphone, thermometer, accelerometer, antenna, etc.) Advantages: → passive networks are highly lin- → electrical signal that controls a process U in ear over a very large dynamic U in U out range and large bandwidths Continuous-time signals: voltage, current, temperature, speed, . . . → analog U out signal-processing cir- Discrete-time signals: daily minimum/maximum temperature, 0 1 / √ LC ω (= 2 πf ) t cuits require little or no power lap intervals in races, sampled continuous signals, . . . → analog circuits cause little ad- � t U in − U out = 1 U out d τ + C d U out ditional interference R L d t Electronics (unlike optics) can only deal easily with time-dependent signals, therefore spatial −∞ signals, such as images, are typically first converted into a time signal with a scanning process (TV, fax, etc.). 2 4
Digital signal processing Objectives Analog/digital and digital/analog converter, CPU, DSP, ASIC, FPGA. By the end of the course, you should be able to → apply basic properties of time-invariant linear systems Advantages: → understand sampling, aliasing, convolution, filtering, the pitfalls of → noise is easy to control after initial quantization spectral estimation → highly linear (within limited dynamic range) → explain the above in time and frequency domain representations → complex algorithms fit into a single chip → use filter-design software → flexibility, parameters can easily be varied in software → visualise and discuss digital filters in the z -domain → use the FFT for convolution, deconvolution, filtering → digital processing is insensitive to component tolerances, aging, → implement, apply and evaluate simple DSP applications in MATLAB environmental conditions, electromagnetic interference → apply transforms that reduce correlation between several signal sources But: → understand and explain limits in human perception that are ex- → discrete-time processing artifacts (aliasing) ploited by lossy compression techniques → can require significantly more power (battery, cooling) → understand the basic principles of several widely-used modulation → digital clock and switching cause interference and audio-visual coding techniques. 5 7 Typical DSP applications Textbooks → R.G. Lyons: Understanding digital signal processing. 3rd ed., → communication systems → astronomy Prentice-Hall, 2010. ( £ 68) modulation/demodulation, channel VLBI, speckle interferometry equalization, echo cancellation → A.V. Oppenheim, R.W. Schafer: Discrete-time signal process- → experimental physics → consumer electronics ing. 3rd ed., Prentice-Hall, 2007. ( £ 47) sensor-data evaluation perceptual coding of audio and video → J. Stein: Digital signal processing – a computer science per- on DVDs, speech synthesis, speech → aviation recognition spective. Wiley, 2000. ( £ 133) → music radar, radio navigation → S.W. Smith: Digital signal processing – a practical guide for synthetic instruments, audio effects, → security engineers and scientists. Newness, 2003. ( £ 48) noise reduction → K. Steiglitz: A digital signal processing primer – with appli- steganography, digital watermarking, → medical diagnostics biometric identification, surveillance cations to digital audio and computer music. Addison-Wesley, systems, signals intelligence, elec- magnetic-resonance and ultrasonic tronic warfare 1996. ( £ 67) imaging, computer tomography, ECG, EEG, MEG, AED, audiology → engineering → Sanjit K. Mitra: Digital signal processing – a computer-based → geophysics approach. McGraw-Hill, 2002. ( £ 38) control systems, feature extraction seismology, oil exploration for pattern recognition 6 8
Units and decibel Sequences and systems A discrete sequence { x n } ∞ Communications engineers often use logarithmic units: n = −∞ is a sequence of numbers → Quantities often vary over many orders of magnitude → difficult . . . , x − 2 , x − 1 , x 0 , x 1 , x 2 , . . . to agree on a common SI prefix (nano, micro, milli, kilo, etc.) where x n denotes the n -th number in the sequence ( n ∈ Z ). A discrete → Quotient of quantities (amplification/attenuation) usually more sequence maps integer numbers onto real (or complex) numbers. interesting than difference We normally abbreviate { x n } ∞ n = −∞ to { x n } , or to { x n } n if the running index is not obvious. The notation is not well standardized. Some authors write x [ n ] instead of x n , others x ( n ). → Signal strength usefully expressed as field quantity (voltage, Where a discrete sequence { x n } samples a continuous function x ( t ) as current, pressure, etc.) or power, but quadratic relationship between these two ( P = U 2 /R = I 2 R ) rather inconvenient x n = x ( t s · n ) = x ( n/f s ) , → Perception is logarithmic (Weber/Fechner law → slide 175) we call t s the sampling period and f s = 1 /t s the sampling frequency . Plus: Using magic special-purpose units has its own odd attractions ( → typographers, navigators) A discrete system T receives as input a sequence { x n } and transforms Neper (Np) denotes the natural logarithm of the quotient of a field it into an output sequence { y n } = T { x n } : quantity F and a reference value F 0 . (rarely used today) discrete Bel (B) denotes the base-10 logarithm of the quotient of a power P . . . , x 2 , x 1 , x 0 , x − 1 , . . . . . . , y 2 , y 1 , y 0 , y − 1 , . . . system T and a reference power P 0 . Common prefix: 10 decibel (dB) = 1 bel. 9 11 Some simple sequences Where P is some power and P 0 a 0 dB reference power, or equally where F is a field quantity and F 0 the corresponding reference level: u n P F Unit-step sequence: 1 10 dB · log 10 = 20 dB · log 10 P 0 F 0 � 0 , n < 0 Common reference values are indicated with suffix after “dB”: u n = 1 , n ≥ 0 0 dBW = 1 W . . . − 3 − 2 − 1 0 1 2 3 . . . n 0 dBm = 1 mW = − 30 dBW 0 dB µ V = 1 µ V δ n 0 dB SPL = 20 µ Pa (sound pressure level) Impulse sequence: 1 0 dB SL = perception threshold (sensation limit) � 1 , n = 0 δ n = Remember: 0 , n � = 0 3 dB = 2 × power, 6 dB = 2 × voltage/pressure/etc. = u n − u n − 1 . . . − 3 − 2 − 1 0 1 2 3 . . . n 10 dB = 10 × power, 20 dB = 10 × voltage/pressure/etc. W.H. Martin: Decibel – the new name for the transmission unit. Bell System Technical Journal, January 1929. 10 12
Sinusoidial sequences Types of discrete systems A cosine wave, frequency f , phase offset θ : A causal system cannot look into the future: x ( t ) = cos(2 π ft + θ ) y n = f ( x n , x n − 1 , x n − 2 , . . . ) Sampling it at sampling rate f s results in the discrete sequence { x n } : A memory-less system depends only on the current input value: x n = cos(2 π fn/f s + θ ) y n = f ( x n ) A delay system shifts a sequence in time: Exercise 1 Use the following MATLAB (or GNU Octave) code to display 41 samples ( ≈ 1 / 200 s = 5 ms) of a 400 Hz sinusoidial wave sampled at y n = x n − d 8 kHz: T is a time-invariant system if for any d n=0:40; fs=8000; f=400; x=cos(2*pi*f*n/fs); stem(n, x); ylim([-1.1 1.1]) { y n } = T { x n } ⇐ ⇒ { y n − d } = T { x n − d } . Try frequencies f of 0, 1000, 2000, 3000, 4000, and 5000 Hz. Also try to T is a linear system if for any pair of sequences { x n } and { x ′ n } negate these frequencies. Do any of these resulting sequences look to be identical, or are they negatives of each other? Also try sin instead of cos . T { a · x n + b · x ′ n } = a · T { x n } + b · T { x ′ n } . Finally, try adding θ phase offsets of ± π / 4, ± π / 2, and ± π . 13 15 Properties of sequences Example: M -point moving average system A sequence { x n } is M − 1 y n = 1 x n − k = x n − M +1 + · · · + x n − 1 + x n � periodic ⇔ ∃ k > 0 : ∀ n ∈ Z : x n = x n + k M M ∞ k =0 � absolutely summable ⇔ | x n | < ∞ It is causal, linear, time-invariant, with memory. With M = 4: n = −∞ ∞ � | x n | 2 square summable ⇔ < ∞ ⇔ “energy signal” x y n = −∞ � �� � “energy ′′ k 1 � | x n | 2 0 < lim < ∞ ⇔ “power signal” 0 1 + 2 k k →∞ n = − k � �� � “average power” This energy/power terminology reflects that if U is a voltage supplied to a load � resistor R , then P = UI = U 2 /R is the power consumed, and P ( t ) d t the energy. It is used even if we drop physical units (e.g., volts) for simplicity in calculations. 14 16
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