Introduction (Vaseghi - Advanced Digital Signal Processing and Noise Reduction), страница 4

At the encoder the gain of each bandis adaptively adjusted to boost low–energy signal components. Dolby A19Applications of Digital Signal ProcessingRelative gain (dB)-25-30-35-40-450.11.010Frequency (kHz)Figure 1.15 Illustration of the pre-emphasis response of Dolby-C: upto 20 dBboost is provided when the signal falls 45 dB below maximum recording level.provides a maximum gain of 10 to 15 dB in each band if the signal levelfalls 45 dB below the maximum recording level. The Dolby B and Dolby Csystems are designed for consumer audio systems, and use two bandsinstead of the four bands used in Dolby A. Dolby B provides a boost of upto 10 dB when the signal level is low (less than 45 dB than the maximumreference) and Dolby C provides a boost of up to 20 dB as illustrated inFigure1.15.1.3.9 Radar Signal Processing: Doppler Frequency ShiftFigure 1.16 shows a simple diagram of a radar system that can be used toestimate the range and speed of an object such as a moving car or a flyingaeroplane.

A radar system consists of a transceiver (transmitter/receiver) thatgenerates and transmits sinusoidal pulses at microwave frequencies. Thesignal travels with the speed of light and is reflected back from any object inits path. The analysis of the received echo provides such information asrange, speed, and acceleration. The received signal has the form20Introductionx ( t ) = A ( t ) cos{ω 0 [ t − 2 r ( t ) / c ]}(1.13)where A(t), the time-varying amplitude of the reflected wave, depends on theposition and the characteristics of the target, r(t) is the time-varying distanceof the object from the radar and c is the velocity of light.

The time-varyingdistance of the object can be expanded in a Taylor series asr ( t ) = r0 + rt +1 2 1 3rt + rt + 2!3!(1.14)where r0 is the distance, r is the velocity, r is the acceleration etc.Approximating r(t) with the first two terms of the Taylor series expansionwe have(1.15)r ( t ) ≈ r0 + rtSubstituting Equation (1.15) in Equation (1.13) yieldsx ( t ) = A ( t ) cos[(ω 0 − 2 rω 0 / c ) t − 2ω 0 r0 / c ](1.16)Note that the frequency of reflected wave is shifted by an amountω d = 2 rω 0 / c(1.17)This shift in frequency is known as the Doppler frequency. If the object ismoving towards the radar then the distance r(t) is decreasing with time, r isnegative, and an increase in the frequency is observed. Conversely if thecos(ω 0t)-2r(t)/c]}Cos{ω 0[tcr=0.5TFigure 1.16 Illustration of a radar system.21Sampling and Analog–to–Digital Conversionobject is moving away from the radar then the distance r(t) is increasing, r ispositive, and a decrease in the frequency is observed.

Thus the frequencyanalysis of the reflected signal can reveal information on the direction andspeed of the object. The distance r0 is given byr0 = 0.5 T × c(1.18)where T is the round-trip time for the signal to hit the object and arrive backat the radar and c is the velocity of light.1.4 Sampling and Analog–to–Digital ConversionA digital signal is a sequence of real–valued or complex–valued numbers,representing the fluctuations of an information bearing quantity with time,space or some other variable. The basic elementary discrete-time signal isthe unit-sample signal δ(m) defined as1δ (m) = 0m=0m≠0(1.19)where m is the discrete time index.

A digital signal x(m) can be expressed asthe sum of a number of amplitude-scaled and time-shifted unit samples asx(m) =∞∑ x(k)δ (m − k)(1.20)k = −∞Figure 1.17 illustrates a discrete-time signal. Many random processes, suchas speech, music, radar and sonar generate signals that are continuous inDiscrete timemFigure 1.17 A discrete-time signal and its envelope of variation with time.22IntroductionAnalog inputy(t)LPF &S/Hya(m)y(m)ADCDigital signalprocessorxa(m)x(m)DACx(t)LPFFigure 1.18 Configuration of a digital signal processing system.time and continuous in amplitude. Continuous signals are termed analogbecause their fluctuations with time are analogous to the variations of thesignal source. For digital processing, analog signals are sampled, and eachsample is converted into an n-bit digit.

The digitisation process should beperformed such that the original signal can be recovered from its digitalversion with no loss of information, and with as high a fidelity as is requiredin an application. Figure 1.18 illustrates a block diagram configuration of adigital signal processor with an analog input. The low-pass filter removesout–of–band signal frequencies above a pre-selected range.

The sample–and–hold (S/H) unit periodically samples the signal to convert thecontinuous-time signal into a discrete-time signal.The analog–to–digital converter (ADC) maps each continuousamplitude sample into an n-bit digit. After processing, the digital output ofthe processor can be converted back into an analog signal using a digital–to–analog converter (DAC) and a low-pass filter as illustrated in Figure 1.18.1.4.1 Time-Domain Sampling and Reconstruction of AnalogSignalsThe conversion of an analog signal to a sequence of n-bit digits consists oftwo basic steps of sampling and quantisation.

The sampling process, whenperformed with sufficiently high speed, can capture the fastest fluctuationsof the signal, and can be a loss-less operation in that the analog signal can berecovered through interpolation of the sampled sequence as described inChapter 10. The quantisation of each sample into an n-bit digit, involvessome irrevocable error and possible loss of information. However, inpractice the quantisation error can be made negligible by using anappropriately high number of bits as in a digital audio hi-fi.

A sampledsignal can be modelled as the product of a continuous-time signal x(t) and aperiodic impulse train p(t) as23Sampling and Analog–to–Digital Conversionxsampled (t ) = x(t ) p (t )∞(1.21)∑ x(t )δ (t − mTs )=m= −∞where Ts is the sampling interval and the sampling function p(t) is definedas∞p( t ) =∑ δ (t − mT s )(1.22)m= − ∞The spectrum P( f ) of the sampling function p(t) is also a periodic impulsetrain given byP( f ) =∞∑ δ ( f − kFs )(1.23)k = −∞where Fs=1/Ts is the sampling frequency. Since multiplication of two timedomain signals is equivalent to the convolution of their frequency spectrawe haveX sampled ( f ) = FT [ x(t ).

p(t )] = X ( f ) * P( f ) =∞∑ δ ( f − kFs )(1.24)k = −∞where the operator FT[.] denotes the Fourier transform. In Equation (1.24)the convolution of a signal spectrum X( f ) with each impulse δ ( f − kFs ) ,shifts X( f ) and centres it on kFs. Hence, as expressed in Equation (1.24),the sampling of a signal x(t) results in a periodic repetition of its spectrumX( f ) centred on frequencies 0, ± Fs , ± 2 Fs , . When the samplingfrequency is higher than twice the maximum frequency content of thesignal, then the repetitions of the signal spectra are separated as shown inFigure 1.19.

In this case, the analog signal can be recovered by passing thesampled signal through an analog low-pass filter with a cut-off frequency ofFs. If the sampling frequency is less than 2Fs, then the adjacent repetitionsof the spectrum overlap and the original spectrum cannot be recovered.

Thedistortion, due to an insufficiently high sampling rate, is irrevocable and isknown as aliasing. This observation is the basis of the Nyquist samplingtheorem which states: a band-limited continuous-time signal, with a highest24IntroductionTime domainFrequency domainx(t)X(f)0B–Btf2B×*Impulse-train-samplingfunction......Ts=xp(t)P( f ) =∞∑ δ ( f − kF )sk = −∞...t0–Fs=Fs=1/TsXp ( f )Impulse-train-sampledsignal...*...–Fs/2tf0×fFs/2SH( f )sh(t)Sample-and-hold function...0=Tst...–FsFs=f|X( f )|Xsh(t)S/H-sampled signal...t...–Fs/20Fs/2fFigure 1.19 Sample-and-Hold signal modelled as impulse-train sampling followedby convolution with a rectangular pulse.frequency content (bandwidth) of B Hz, can be recovered from its samplesprovided that the sampling speed Fs>2B samples per second.In practice sampling is achieved using an electronic switch that allows acapacitor to charge up or down to the level of the input voltage once everyTs seconds as illustrated in Figure 1.20.

The sample-and-hold signal can bemodelled as the output of a filter with a rectangular impulse response, andwith the impulse–train–sampled signal as the input as illustrated inFigure1.19.25Sampling and Analog–to–Digital ConversionR2x(t)Rx(mT s )CTsFigure 1.20 A simplified sample-and-hold circuit diagram.1.4.2 QuantisationFor digital signal processing, continuous-amplitude samples from thesample-and-hold are quantised and mapped into n-bit binary digits. Forquantisation to n bits, the amplitude range of the signal is divided into 2ndiscrete levels, and each sample is quantised to the nearest quantisationlevel, and then mapped to the binary code assigned to that level. Figure 1.21illustrates the quantisation of a signal into 4 discrete levels. Quantisation is amany-to-one mapping, in that all the values that fall within the continuum ofa quantisation band are mapped to the centre of the band. The mappingbetween an analog sample xa(m) and its quantised value x(m) can beexpressed asx(m) = Q[x a (m)](1.25)where Q[· ] is the quantising function.The performance of a quantiser is measured by signal–to–quantisationnoise ratio SQNR per bit.