Introduction (Vaseghi - Advanced Digital Signal Processing and Noise Reduction), страница 3

A linear predictor forecasts the amplitude of the signal attime m, x(m) , using a linear combination of P previous samples[x ( m − 1),, x (m − P )] asPxˆ (m) = ∑ a k x(m − k )(1.3)k =1where xˆ (m) is the prediction of the signal x(m) , and the vectora T = [a1 , , a P ] is the coefficients vector of a predictor of order P. ThePitch periodRandomsourceExcitationGlottal (pitch)modelP(z)Vocal tractmodelH(z)Figure 1.8 Linear predictive model of speech.Speech12Introductionx(m)e(m)u(m)Ga2aP–1x(m–P)z...x(m-2)a1z –1–1x(m-1)zFigure 1.9 Illustration of a signal generated by an all-pole, linear predictionmodel.prediction error e( m), i.e. the difference between the actual sample x (m)and its predicted value xˆ (m) , is defined asPe(m) = x(m) − ∑ ak x(m − k)(1.4)k =1The prediction error e(m) may also be interpreted as the random excitationor the so-called innovation content of x(m) .

From Equation (1.4) a signalgenerated by a linear predictor can be synthesised asPx (m) = ∑ ak x (m − k ) + e (m)(1.5)k =1Equation (1.5) describes a speech synthesis model illustrated in Figure 1.9.1.3.5 Digital Coding of Audio SignalsIn digital audio, the memory required to record a signal, the bandwidthrequired for signal transmission and the signal–to–quantisation–noise ratioare all directly proportional to the number of bits per sample. The objectivein the design of a coder is to achieve high fidelity with as few bits persample as possible, at an affordable implementation cost.

Audio signalcoding schemes utilise the statistical structures of the signal, and a model ofthe signal generation, together with information on the psychoacoustics andthe masking effects of hearing. In general, there are two main categories ofaudio coders: model-based coders, used for low–bit–rate speech coding in13Applications of Digital Signal ProcessingSpeech x(m)Pitch and vocal-tractSynthesisercoefficientscoefficientsScalarquantiserModel-basedExcitation addressspeech analysis Excitation e(m)Vectorquantiser(a) Source coderPitch coefficientsExcitationaddressExcitationcodebookPitch filterVocal-tract coefficientsVocal-tract filterReconstructedspeech(b) Source decoderFigure 1.10 Block diagram configuration of a model-based speech coder.applications such as cellular telephony; and transform-based coders used inhigh–quality coding of speech and digital hi-fi audio.Figure 1.10 shows a simplified block diagram configuration of a speechcoder–synthesiser of the type used in digital cellular telephone.

The speechsignal is modelled as the output of a filter excited by a random signal. Therandom excitation models the air exhaled through the lung, and the filtermodels the vibrations of the glottal cords and the vocal tract. At thetransmitter, speech is segmented into blocks of about 30 ms long duringwhich speech parameters can be assumed to be stationary. Each block ofspeech samples is analysed to extract and transmit a set of excitation andfilter parameters that can be used to synthesis the speech.

At the receiver, themodel parameters and the excitation are used to reconstruct the speech.A transform-based coder is shown in Figure 1.11. The aim oftransformation is to convert the signal into a form where it lends itself to amore convenient and useful interpretation and manipulation. In Figure 1.11the input signal is transformed to the frequency domain using a filter bank,or a discrete Fourier transform, or a discrete cosine transform. Three mainadvantages of coding a signal in the frequency domain are:(a) The frequency spectrum of a signal has a relatively well–definedstructure, for example most of the signal power is usuallyconcentrated in the lower regions of the spectrum.14IntroductionBinary coded signal^X(0)x(1)X(1)n 1 bps^X(1)x(2)...x(N-1)X(2)...X(N-1)n 2 bps...n N-1 bpsDecodern 0 bpsEncoderX(0)Transform Tx(0)^X(2)...^X(N-1)Inverse Transform T-1Input signalReconstructedsignal^x(0)^x(1)^x(2)...^x(N-1)Figure 1.11 Illustration of a transform-based coder.(b) A relatively low–amplitude frequency would be masked in the nearvicinity of a large–amplitude frequency and can therefore becoarsely encoded without any audible degradation.(c) The frequency samples are orthogonal and can be codedindependently with different precisions.The number of bits assigned to each frequency of a signal is a variablethat reflects the contribution of that frequency to the reproduction of aperceptually high quality signal.

In an adaptive coder, the allocation of bitsto different frequencies is made to vary with the time variations of thepower spectrum of the signal.1.3.6 Detection of Signals in NoiseIn the detection of signals in noise, the aim is to determine if the observationconsists of noise alone, or if it contains a signal. The noisy observationy( m) can be modelled asy(m) = b(m)x(m) + n(m)(1.6)where x(m) is the signal to be detected, n(m) is the noise and b(m) is abinary-valued state indicator sequence such that b(m) = 1 indicates thepresence of the signal x (m) and b( m) = 0 indicates that the signal is absent.If the signal x(m) has a known shape, then a correlator or a matched filter15Applications of Digital Signal Processingy(m)=x(m)+n(m)z(m)Matched filterh(m) = x(N – 1–m)Thresholdcomparator^b(m)Figure 1.12 Configuration of a matched filter followed by a threshold comparator fordetection of signals in noise.can be used to detect the signal as shown in Figure 1.12.

The impulseresponse h(m) of the matched filter for detection of a signal x(m) is thetime-reversed version of x (m) given byh ( m) = x ( N − 1 − m)0 ≤ m≤ N −1(1.7)where N is the length of x(m) . The output of the matched filter is given byN −1z ( m ) = ∑ h ( m − k ) y ( m)(1.8)m=0The matched filter output is compared with a threshold and a binarydecision is made as1 if z (m) ≥ thresholdbˆ(m) = (1.9) 0 otherwisewhere bˆ (m) is an estimate of the binary state indicator sequence b(m), andit may be erroneous in particular if the signal–to–noise ratio is low.

Table1.1lists four possible outcomes that together b(m) and its estimate bˆ (m) canassume. The choice of the threshold level affects the sensitivity of thebˆ (m)0011b(m)0101Detector decisionSignal absent CorrectSignal absent(Missed)Signal present (False alarm)Signal present CorrectTable 1.1 Four possible outcomes in a signal detection problem.16IntroductionFigure 1.13 Sonar: detection of objects using the intensity and time delay ofreflected sound waves.detector.

The higher the threshold, the less the likelihood that noise wouldbe classified as signal, so the false alarm rate falls, but the probability ofmisclassification of signal as noise increases. The risk in choosing athreshold value θ can be expressed asR (Threshold = θ )= PFalse Alarm (θ ) + PMiss (θ )(1.10)The choice of the threshold reflects a trade-off between the misclassificationrate PMiss(θ) and the false alarm rate PFalse Alarm(θ).1.3.7 Directional Reception of Waves: Beam-formingBeam-forming is the spatial processing of plane waves received by an arrayof sensors such that the waves incident at a particular spatial angle arepassed through, whereas those arriving from other directions are attenuated.Beam-forming is used in radar and sonar signal processing (Figure 1.13) tosteer the reception of signals towards a desired direction, and in speechprocessing for reducing the effects of ambient noise.To explain the process of beam-forming consider a uniform linear arrayof sensors as illustrated in Figure 1.14.

The term linear array implies thatthe array of sensors is spatially arranged in a straight line and with equalspacing d between the sensors. Consider a sinusoidal far–field plane wavewith a frequency F0 propagating towards the sensors at an incidence angleof θ as illustrated in Figure 1.14. The array of sensors samples the incoming17Applications of Digital Signal Processingwave as it propagates in space.

The time delay for the wave to travel adistance of d between two adjacent sensors is given byτ=d sin θc(1.11)where c is the speed of propagation of the wave in the medium. The phasedifference corresponding to a delay of τ is given byϕ = 2πd sin θτ= 2πF0T0c(1.12)where T0 is the period of the sine wave. By inserting appropriate correctiveArray of sensorsArray of filters0...z –1z –1W1,1W1,0θ dW1,P–1Incident planewave+1...z –1d sinθz –1W2,1W 2,0W2,P–1+Output......N-1...z –1WN–1,0θz –1WN–1,P–1WN–1,1+Figure 1.14 Illustration of a beam-former, for directional reception of signals.18Introductiontime delays in the path of the samples at each sensor, and then averaging theoutputs of the sensors, the signals arriving from the direction θ will be timealigned and coherently combined, whereas those arriving from otherdirections will suffer cancellations and attenuations.

Figure 1.14 illustrates abeam-former as an array of digital filters arranged in space. The filter arrayacts as a two–dimensional space–time signal processing system. The spacefiltering allows the beam-former to be steered towards a desired direction,for example towards the direction along which the incoming signal has themaximum intensity. The phase of each filter controls the time delay, and canbe adjusted to coherently combine the signals.

The magnitude frequencyresponse of each filter can be used to remove the out–of–band noise.1.3.8 Dolby Noise ReductionDolby noise reduction systems work by boosting the energy and the signalto noise ratio of the high–frequency spectrum of audio signals. The energyof audio signals is mostly concentrated in the low–frequency part of thespectrum (below 2 kHz). The higher frequencies that convey quality andsensation have relatively low energy, and can be degraded even by a lowamount of noise. For example when a signal is recorded on a magnetic tape,the tape “hiss” noise affects the quality of the recorded signal.

On playback,the higher–frequency part of an audio signal recorded on a tape have smallersignal–to–noise ratio than the low–frequency parts. Therefore noise at highfrequencies is more audible and less masked by the signal energy. Dolbynoise reduction systems broadly work on the principle of emphasising andboosting the low energy of the high–frequency signal components prior torecording the signal. When a signal is recorded it is processed and encodedusing a combination of a pre-emphasis filter and dynamic rangecompression. At playback, the signal is recovered using a decoder based ona combination of a de-emphasis filter and a decompression circuit.

Theencoder and decoder must be well matched and cancel out each other inorder to avoid processing distortion.Dolby has developed a number of noise reduction systems designatedDolby A, Dolby B and Dolby C. These differ mainly in the number of bandsand the pre-emphasis strategy that that they employ. Dolby A, developed forprofessional use, divides the signal spectrum into four frequency bands:band 1 is low-pass and covers 0 Hz to 80 Hz; band 2 is band-pass and covers80 Hz to 3 kHz; band 3 is high-pass and covers above 3 kHz; and band 4 isalso high-pass and covers above 9 kHz.