Noise and Distortion (Vaseghi - Advanced Digital Signal Processing and Noise Reduction)

Advanced Digital Signal Processing and Noise Reduction, Second Edition.Saeed V. VaseghiCopyright © 2000 John Wiley & Sons LtdISBNs: 0-471-62692-9 (Hardback): 0-470-84162-1 (Electronic)2NOISE AND DISTORTION2.1 Introduction2.2 White Noise2.3 Coloured Noise2.4 Impulsive Noise2.5 Transient Noise PulsesN2.62.72.82.92.10Thermal NoiseShot NoiseElectromagnetic NoiseChannel DistortionsModelling Noiseoise can be defined as an unwanted signal that interferes with thecommunication or measurement of another signal.

A noise itself is asignal that conveys information regarding the source of the noise.For example, the noise from a car engine conveys information regarding thestate of the engine. The sources of noise are many, and vary from audiofrequency acoustic noise emanating from moving, vibrating or collidingsources such as revolving machines, moving vehicles, computer fans,keyboard clicks, wind, rain, etc.

to radio-frequency electromagnetic noisethat can interfere with the transmission and reception of voice, image anddata over the radio-frequency spectrum. Signal distortion is the term oftenused to describe a systematic undesirable change in a signal and refers tochanges in a signal due to the non–ideal characteristics of the transmissionchannel, reverberations, echo and missing samples.Noise and distortion are the main limiting factors in communication andmeasurement systems. Therefore the modelling and removal of the effects ofnoise and distortion have been at the core of the theory and practice ofcommunications and signal processing.

Noise reduction and distortionremoval are important problems in applications such as cellular mobilecommunication, speech recognition, image processing, medical signalprocessing, radar, sonar, and in any application where the signals cannot beisolated from noise and distortion. In this chapter, we study thecharacteristics and modelling of several different forms of noise.30Noise and Distortion2.1 IntroductionNoise may be defined as any unwanted signal that interferes with thecommunication, measurement or processing of an information-bearingsignal. Noise is present in various degrees in almost all environments.

Forexample, in a digital cellular mobile telephone system, there may be severalvariety of noise that could degrade the quality of communication, such asacoustic background noise, thermal noise, electromagnetic radio-frequencynoise, co-channel interference, radio-channel distortion, echo and processingnoise. Noise can cause transmission errors and may even disrupt acommunication process; hence noise processing is an important part ofmodern telecommunication and signal processing systems. The success of anoise processing method depends on its ability to characterise and model thenoise process, and to use the noise characteristics advantageously todifferentiate the signal from the noise. Depending on its source, a noise canbe classified into a number of categories, indicating the broad physicalnature of the noise, as follows:(a) Acoustic noise: emanates from moving, vibrating, or collidingsources and is the most familiar type of noise present in variousdegrees in everyday environments.

Acoustic noise is generated bysuch sources as moving cars, air-conditioners, computer fans, traffic,people talking in the background, wind, rain, etc.(b) Electromagnetic noise: present at all frequencies and in particular atthe radio frequencies. All electric devices, such as radio andtelevision transmitters and receivers, generate electromagnetic noise.(c) Electrostatic noise: generated by the presence of a voltage with orwithout current flow. Fluorescent lighting is one of the morecommon sources of electrostatic noise.(d) Channel distortions, echo, and fading: due to non-idealcharacteristics of communication channels. Radio channels, such asthose at microwave frequencies used by cellular mobile phoneoperators, are particularly sensitive to the propagation characteristicsof the channel environment.(e) Processing noise: the noise that results from the digital/analogprocessing of signals, e.g.

quantisation noise in digital coding ofspeech or image signals, or lost data packets in digital datacommunication systems.31White NoiseDepending on its frequency or time characteristics, a noise process canbe classified into one of several categories as follows:(a) Narrowband noise: a noise process with a narrow bandwidth such asa 50/60 Hz ‘hum’ from the electricity supply.(b) White noise: purely random noise that has a flat power spectrum.White noise theoretically contains all frequencies in equal intensity.(c) Band-limited white noise: a noise with a flat spectrum and a limitedbandwidth that usually covers the limited spectrum of the device orthe signal of interest.(d) Coloured noise: non-white noise or any wideband noise whosespectrum has a non-flat shape; examples are pink noise, brown noiseand autoregressive noise.(e) Impulsive noise: consists of short-duration pulses of randomamplitude and random duration.(f) Transient noise pulses: consists of relatively long duration noisepulses.2.2 White NoiseWhite noise is defined as an uncorrelated noise process with equal power atall frequencies (Figure 2.1).

A noise that has the same power at allfrequencies in the range of ±∞ would necessarily need to have infinitepower, and is therefore only a theoretical concept. However a band-limitednoise process, with a flat spectrum covering the frequency range of a bandlimited communication system, is to all intents and purposes from the pointof view of the system a white noise process.

For example, for an audiosystem with a bandwidth of 10 kHz, any flat-spectrum audio noise with abandwidth greater than 10 kHz looks like a white noise.Pnn(k)rnn(k)210-1-2050100150200250300mk(a)(b)(c)Figure 2.1 Illustration of (a) white noise, (b) its autocorrelation, and(c) its power spectrum.f32Noise and DistortionThe autocorrelation function of a continuous-time zero-mean white noiseprocess with a variance of σ 2 is a delta function given byrNN (τ ) = E [ N (t ) N (t + τ )] = σ 2δ (τ )(2.1)The power spectrum of a white noise, obtained by taking the Fouriertransform of Equation (2.1), is given by∞PNN ( f ) = ∫ rNN (t )e − j 2πft dt = σ 2(2.2)−∞Equation (2.2) shows that a white noise has a constant power spectrum.A pure white noise is a theoretical concept, since it would need to haveinfinite power to cover an infinite range of frequencies.

Furthermore, adiscrete-time signal by necessity has to be band-limited, with its highestfrequency less than half the sampling rate. A more practical concept is bandlimited white noise, defined as a noise with a flat spectrum in a limitedbandwidth. The spectrum of band-limited white noise with a bandwidth of BHz is given byσ 2 ,PNN ( f ) = 0,| f |≤ Botherwise(2.3)Thus the total power of a band-limited white noise process is 2B σ 2 . Theautocorrelation function of a discrete-time band-limited white noise processis given bysin( 2πBTs k )rNN (Ts k ) = 2 Bσ 2(2.4)2πBTs kwhere Ts is the sampling period.

For convenience of notation Ts is usuallyassumed to be unity. For the case when Ts=1/2B, i.e. when the sampling rateis equal to the Nyquist rate, Equation (2.4) becomesrNN (Ts k ) = 2 Bσ 2sin (πk )= 2 Bσ 2δ (k )πkIn Equation (2.5) the autocorrelation function is a delta function.(2.5)33Coloured Noise2.3 Coloured NoiseAlthough the concept of white noise provides a reasonably realistic andmathematically convenient and useful approximation to some predominantnoise processes encountered in telecommunication systems, many othernoise processes are non-white.

The term coloured noise refers to anybroadband noise with a non-white spectrum. For example most audiofrequency noise, such as the noise from moving cars, noise from computerfans, electric drill noise and people talking in the background, has a nonwhite predominantly low-frequency spectrum. Also, a white noise passingthrough a channel is “coloured” by the shape of the channel spectrum. Twoclassic varieties of coloured noise are so-called pink noise and brown noise,shown in Figures 2.2 and 2.3.0Magnitude dBx(m)m– 300FrequencyFs /2(a)(b)Figure 2.2 (a) A pink noise signal and (b) its magnitude spectrum.0Magnitude dBx(m)m– 50Frequency(a)(b)Figure 2.3 (a) A brown noise signal and (b) its magnitude spectrum.Fs /2Noise and Distortion342.4 Impulsive NoiseImpulsive noise consists of short-duration “on/off” noise pulses, caused by avariety of sources, such as switching noise, adverse channel environment ina communication system, drop-outs or surface degradation of audiorecordings, clicks from computer keyboards, etc.

Figure 2.4(a) shows anideal impulse and its frequency spectrum. In communication systems, a realimpulsive-type noise has a duration that is normally more than one samplelong. For example, in the context of audio signals, short-duration, sharppulses, of up to 3 milliseconds (60 samples at a 20 kHz sampling rate) maybe considered as impulsive noise. Figures 2.4(b) and (c) illustrate twoexamples of short-duration pulses and their respective spectra.In a communication system, an impulsive noise originates at some pointin time and space, and then propagates through the channel to the receiver.The received noise is time-dispersed and shaped by the channel, and can beconsidered as the channel impulse response. In general, the characteristics ofa communication channel may be linear or non-linear, stationary or timevarying.

Furthermore, many communication systems, in response to a largeamplitude impulse, exhibit a non-linear characteristic.ni1(m) =δ (m)Ni1 (f)⇔(a)mfni2(m)Ni2 (f)⇔(b)mfni3(m)Ni3 (f)⇔(c)mfFigure 2.4 Time and frequency sketches of: (a) an ideal impulse, (b) and (c) shortduration pulses.35Transient Noise Pulsesni1(m)ni2(m)mm(a)ni3(m)m(c)(b)Figure 2.5 Illustration of variations of the impulse response of a non-linear systemwith the increasing amplitude of the impulse.Figure 2.5 illustrates some examples of impulsive noise, typical ofthose observed on an old gramophone recording. In this case, thecommunication channel is the playback system, and may be assumed to betime-invariant. The figure also shows some variations of the channelcharacteristics with the amplitude of impulsive noise. For example, inFigure 2.5(c) a large impulse excitation has generated a decaying transientpulse.