Biosignal and Biomedical Image Processing MATLAB-Based Applications Muya phần 2 docx

16 Chapter 1 the downward slope (sometimes referred to as the rolloff ) is increased by 20 db/decade. Figure 1.8 shows the frequency plot of a second-order (two-pole with a slope of 40 db/decade) and a 12th-order lowpass filter, both having the same cutoff frequency, f c , and hence, the same bandwidth. The steeper slope or rolloff of the 12-pole filter is apparent. In principle, a 12-pole lowpass filter would have a slope of 240 db/decade (12 × 20 db/decade). In fact, this fre- quency characteristic is theoretical because in real analog filters parasitic com- ponents and inaccuracies in the circuit elements limit the actual attenuation that can be obtained. The same rationale applies to highpass filters except that the frequency plot decreases with decreasing frequency at a rate of 20 db/decade for each highpass filter pole. Filter Initial Sharpness As shown in Figure 1.8, both the slope and the initial sharpness increase with filter order (number of poles), but increasing filter order also increases the com- F IGURE 1.8 Frequency plot of a second-order (2-pole) and a 12th-order lowpass filter with the same cutoff frequency. The higher order filter more closely ap- proaches the sharpness of an ideal filter. TLFeBOOK Introduction 17 plexity, hence the cost, of the filter. It is possible to increase the initial sharpness of the filter’s attenuation characteristics without increasing the order of the filter, if you are willing to except some unevenness, or ripple, in the passband. Figure 1.9 shows two lowpass, 4 th -order filters, differing in the initial sharpness of the attenuation. The one marked Butterworth has a smooth passband, but the initial attenuation is not as sharp as the one marked Chebychev; which has a passband that contains ripples. This property of analog filters is also seen in digital filters and will be discussed in detail in Chapter 4. F IGURE 1.9 Two filters having the same order (4-pole) and cutoff frequency, but differing in the sharpness of the initial slope. The filter marked Chebychev has a steeper initial slope or rolloff, but contains ripples in the passband. TLFeBOOK 18 Chapter 1 ANALOG-TO-DIGITAL CONVERSION: BASIC CONCEPTS The last analog element in a typical measurement system is the analog-to-digital converter (ADC), Figure 1.1. As the name implies, this electronic component converts an analog voltage to an equivalent digital number. In the process of analog-to-digital conversion an analog or continuous waveform, x (t), is con- verted into a discrete waveform, x(n), a function of real numbers that are defined only at discrete integers, n. To convert a continuous waveform to digital format requires slicing the signal in two ways: slicing in time and slicing in amplitude (Figure 1.10). Slicing the signal into discrete points in time is termed time sampling or simply sampling. Time slicing samples the continuous waveform, x(t), at dis- crete prints in time, nT s , where T s is the sample interval. The consequences of time slicing are discussed in the next chapter. The same concept can be applied to images wherein a continuous image such as a photograph that has intensities that vary continuously across spatial distance is sampled at distances of S mm. In this case, the digital representation of the image is a two-dimensional array. The consequences of spatial sampling are discussed in Chapter 11. Since the binary output of the ADC is a discrete integer while the analog signal has a continuous range of values, analog-to-digital conversion also re- quires the analog signal to be sliced into discrete levels, a process termed quanti- zation, Figure 1.10. The equivalent number can only approximate the level of F IGURE 1.10 Converting a continuous signal (solid line) to discrete format re- quires slicing the signal in time and amplitude. The result is a series of discrete points (X’s) that approximate the original signal. TLFeBOOK Introduction 19 the analog signal, and the degree of approximation will depend on the range of binary numbers and the amplitude of the analog signal. For example, if the output of the ADC is an 8-bit binary number capable of 2 8 or 256 discrete states, and the input amplitude range is 0.0–5.0 volts, then the quantization interval will be 5/256 or 0.0195 volts. If, as is usually the case, the analog signal is time varying in a continuous manner, it must be approximated by a series of binary numbers representing the approximate analog signal level at discrete points in time (Figure 1.10). The errors associated with amplitude slicing, or quantization, are described in the next section, and the potential error due to sampling is covered in Chapter 2. The remainder of this section briefly describes the hard- ware used to achieve this approximate conversion. Analog-to-Digital Conversion Techniques Various conversion rules have been used, but the most common is to convert the voltage into a proportional binary number. Different approaches can be used to implement the conversion electronically; the most common is the successive approximation technique described at the end of this section. ADC’s differ in conversion range, speed of conversion, and resolution. The range of analog volt- ages that can be converted is frequently software selectable, and may, or may not, include negative voltages. Typical ranges are from 0.0–10.0 volts or less, or if negative values are possible ± 5.0 volts or less. The speed of conversion is specified in terms of samples per second, or conversion time. For example, an ADC with a conversion time of 10 µsec should, logically, be able to operate at up to 100,000 samples per second (or simply 100 kHz). Typical conversion rates run up to 500 kHz for moderate cost converters, but off-the-shelf converters can be obtained with rates up to 10–20 MHz. Except for image processing systems, lower conversion rates are usually acceptable for biological signals. Even image processing systems may use downsampling techniques to reduce the required ADC conversion rate and, hence, the cost. A typical ADC system involves several components in addition to the actual ADC element, as shown in Figure 1.11. The first element is an N-to-1 analog switch that allows multiple input channels to be converted. Typical ADC systems provide up to 8 to 16 channels, and the switching is usually software- selectable. Since a single ADC is doing the conversion for all channels, the conversion rate for any given channel is reduced in proportion to the number of channels being converted. Hence, an ADC system with converter element that had a conversion rate of 50 kHz would be able to sample each of eight channels at a theoretical maximum rate of 50/8 = 6.25 kHz. The Sample and Hold is a high-speed switch that momentarily records the input signal, and retains that signal value at its output. The time the switch is closed is termed the aperture time. Typical values range around 150 ns, and, except for very fast signals, can be considered basically instantaneous. This TLFeBOOK 20 Chapter 1 F IGURE 1.11 Block diagram of a typical analog-to-digital conversion system. instantaneously sampled voltage value is held (as a charge on a capacitor) while the ADC element determines the equivalent binary number. Again, it is the ADC element that determines the overall speed of the conversion process. Quantization Error Resolution is given in terms of the number of bits in the binary output with the assumption that the least significant bit (LSB) in the output is accurate (which may not always be true). Typical converters feature 8-, 12-, and 16-bit output with 12 bits presenting a good compromise between conversion resolution and cost. In fact, most signals do not have a sufficient signal-to-noise ratio to justify a higher resolution; you are simply obtaining a more accurate conversion of the noise. For example, assuming that converter resolution is equivalent to the LSB, then the minimum voltage that can be resolved is the same as the quantization voltage described above: the voltage range divided by 2 N , where N is the number of bits in the binary output. The resolution of a 5-volt, 12-bit ADC is 5.0/2 12 = 5/4096 = 0.0012 volts. The dynamic range of a 12-bit ADC, the range from the smallest to the largest voltage it can convert, is from 0.0012 to 5 volts: in db this is 20 * log*10 12 * = 167 db. Since typical signals, especially those of biologi- cal origin, have dynamic ranges rarely exceeding 60 to 80 db, a 12-bit converter with the dynamic range of 167 db may appear to be overkill. However, having this extra resolution means that not all of the range need be used, and since 12- bit ADC’s are only marginally more expensive than 8-bit ADC’s they are often used even when an 8-bit ADC (with dynamic range of over 100 DB, would be adequate). A 12-bit output does require two bytes to store and will double the memory requirements over an 8-bit ADC. TLFeBOOK Introduction 21 The number of bits used for conversion sets a lower limit on the resolu- tion, and also determines the quantization error (Figure 1.12). This error can be thought of as a noise process added to the signal. If a sufficient number of quantization levels exist (say N > 64), the distortion produced by quantization error may be modeled as additive independent white noise with zero mean with the variance determined by the quantization step size, δ=V MAX /2 N . Assuming that the error is uniformly distributed between −δ/2 +δ/2, the variance, σ,is: σ= ∫ δ/2 −δ/2 η 2 /δ dη=V 2 Max (2 −2N )/12 (8) Assuming a uniform distribution, the RMS value of the noise would be just twice the standard deviation, σ. Further Study: Successive Approximation The most popular analog-to-digital converters use a rather roundabout strategy to find the binary number most equivalent to the input analog voltage—a digi- tal-to-analog converter (DAC) is placed in a feedback loop. As shown Figure 1.13, an initial binary number stored in the buffer is fed to a DAC to produce a F IGURE 1.12 Quantization (amplitude slicing) of a continuous waveform. The lower trace shows the error between the quantized signal and the input. TLFeBOOK 22 Chapter 1 F IGURE 1.13 Block diagram of an analog-to-digital converter. The input analog voltage is compared with the output of a digital-to-analog converter. When the two voltages match, the number held in the binary buffer is equivalent to the input voltage with the resolution of the converter. Different strategies can be used to adjust the contents of the binary buffer to attain a match. proportional voltage, V DAC . This DAC voltage, V DAC , is then compared to the input voltage, and the binary number in the buffer is adjusted until the desired level of match between V DAC and V in is obtained. This approach begs the question “How are DAC’s constructed?” In fact, DAC’s are relatively easy to construct using a simple ladder network and the principal of current superposition. The controller adjusts the binary number based on whether or not the comparator finds the voltage out of the DAC, V DAC , to be greater or less than the input voltage, V in . One simple adjustment strategy is to increase the binary number by one each cycle if V DAC < V in , or decrease it otherwise. This so-called tracking ADC is very fast when V in changes slowly, but can take many cycles when V in changes abruptly (Figure 1.14). Not only can the conversion time be quite long, but it is variable since it depends on the dynamics of the input signal. This strategy would not easily allow for sampling an analog signal at a fixed rate due to the variability in conversion time. An alternative strategy termed successive approximation allows the con- version to be done at a fixed rate and is well-suited to digital technology. The successive approximation strategy always takes the same number of cycles irre- spective of the input voltage. In the first cycle, the controller sets the most significant bit (MSB) of the buffer to 1; all others are cleared. This binary number is half the maximum possible value (which occurs when all the bits are TLFeBOOK Introduction 23 F IGURE 1.14 Voltage waveform of an ADC that uses a tracking strategy. The ADC voltage (solid line) follows the input voltage (dashed line) fairly closely when the input voltage varies slowly, but takes many cycles to “catch up” to an abrupt change in input voltage. 1), so the DAC should output a voltage that is half its maximum voltage—that is, a voltage in the middle of its range. If the comparator tells the controller that V in > V DAC , then the input voltage, V in , must be greater than half the maximum range, and the MSB is left set. If V in < V DAC , then that the input voltage is in the lower half of the range and the MSB is cleared (Figure 1.15). In the next cycle, the next most significant bit is set, and the same comparison is made and the same bit adjustment takes place based on the results of the comparison (Figure 1.15). After N cycles, where N is the number of bits in the digital output, the voltage from the DAC, V DAC , converges to the best possible fit to the input voltage, V in . Since V in Ϸ V DAC , the number in the buffer, which is proportional to V DAC , is the best representation of the analog input voltage within the resolu- tion of the converter. To signal the end of the conversion process, the ADC puts TLFeBOOK 24 Chapter 1 F IGURE 1.15 V in and V DAC in a 6-bit ADC using the successive approximation strategy. In the first cycle, the MSB is set (solid line) since V in > V DAC . In the next two cycles, the bit being tested is cleared because V in < V DAC when this bit was set. For the fourth and fifth cycles the bit being tested remained set and for the last cycle it was cleared. At the end of the sixth cycle a conversion complete flag is set to signify the end of the conversion process. out a digital signal or flag indicating that the conversion is complete (Figure 1.15). TIME SAMPLING: BASICS Time sampling transforms a continuous analog signal into a discrete time signal, a sequence of numbers denoted as x(n) = [x 1 , x 2 , x 3 , x N ],* Figure 1.16 (lower trace). Such a representation can be thought of as an array in computer memory. (It can also be viewed as a vector as shown in the next chapter.) Note that the array position indicates a relative position in time, but to relate this number sequence back to an absolute time both the sampling interval and sampling onset time must be known. However, if only the time relative to conversion onset is important, as is frequently the case, then only the sampling interval needs to be *In many textbooks brackets, [ ], are used to denote digitized variables; i.e., x[n]. Throughout this text we reserve brackets to indicate a series of numbers, or vector, following the MATLAB format. TLFeBOOK Introduction 25 F IGURE 1.16 A continuous signal (upper trace) is sampled at discrete points in time and stored in memory as an array of proportional numbers (lower trace). known. Converting back to relative time is then achieved by multiplying the sequence number, n, by the sampling interval, T s : x(t) = x(nT s ). Sampling theory is discussed in the next chapter and states that a sinusoid can be uniquely reconstructed providing it has been sampled by at least two equally spaced points over a cycle. Since Fourier series analysis implies that any signal can be represented is a series of sin waves (see Chapter 3), then by extension, a signal can be uniquely reconstructed providing the sampling fre- quency is twice that of the highest frequency in the signal. Note that this highest frequency component may come from a noise source and could be well above the frequencies of interest. The inverse of this rule is that any signal that con- tains frequency components greater than twice the sampling frequency cannot be reconstructed, and, hence, its digital representation is in error. Since this error is introduced by undersampling, it is inherent in the digital representation and no amount of digital signal processing can correct this error. The specific nature of this under-sampling error is termed aliasing and is described in a discussion of the consequences of sampling in Chapter 2. From a practical standpoint, aliasing must be avoided either by the use of very high sampling rates—rates that are well above the bandwidth of the analog system—or by filtering the analog signal before analog-to-digital conversion. Since extensive sampling rates have an associated cost, both in terms of the TLFeBOOK [...]... autocorrelation curves shown in Figure 2. 8 above The two figures were obtained for bandlimited noise having bandwidths of π /20 rad/sec and π/8 rad/sec The variable wc specifies the cutoff frequency of the lowpass filter in the code below The theory and implementation of a lowpass filter such as used below are presented in Chapter 4 Example 2. 2 Generate bandlimited noise and compute and plot the autocorrelation... to avoid the value judgement implied by the word “normal.” 31 TLFeBOOK 32 Chapter 2 FIGURE 2. 1 (A) The distribution of 20 ,000 uniformly distributed random numbers (B) The distribution of 20 ,000 numbers, each of which is the average of two uniformly distributed random numbers (C) and (D) The distribution obtained when 3 and 8 random numbers, still uniformly distributed, are averaged together Although... input a series of sine waves into the box and record the following output: Frequency (Hz): 2 10 20 60 100 125 150 20 0 300 400 Vout volts rms: 15×10−7 0.1×10−3 0.0 02 0 .2 1.5 3 .28 4.47 4.97 4.99 5.0 What is the cutoff frequency and order of this filter? 7 An 8-bit ADC converter that has an input range of ± 5 volts is used to convert a signal that varies between ± 2 volts What is the SNR of the input if... and compute and plot the autocorrelation function for two different bandwidths % Example 2. 2 and Figure 2. 8 % Generate colored noise having two different bandwidths % and evaluate using autocorrelation % close all; clear all; N = 1 024 ; % Size of arrays L = 100; % FIR filter length w = pi /20 ; % Lowpass filter cutoff frequency noise = randn(N,1); % Generate noise % % Compute the impulse response of a lowpass... crosscorrelation and autocorrelation sequences are derived from the expectation operation applied to infinite data In signal processing, data lengths are finite, so the expec- FIGURE 2. 8 Autocorrelation functions of a random time series with a narrow bandwidth (left) and broader bandwidth (right) Note the inverse relationship between the autocorrelation function and the spectrum: the broader the bandwidth... mean, or average value, and its variance, the term 2 in Eq (1) The arithmetic quantities of mean and variance are frequently used in signal processing algorithms, and their computation is well-suited to discrete data The mean value of a discrete array of N samples is evaluated as: x= ¯ 1 N N ∑x (2) k k=1 Note that the summation in Eq (2) is made between 1 and N as opposed to 0 and N − 1 This protocol... standard deviation is the square root of the variance and the standard deviations add as the √N times the average standard deviation [Eq (5)] Accordingly, the mean standard deviation is the average of the individual standard deviations divided by √N [Eq (6)] From Eq (4): N ∑ σ 2k, k=1 N hence: ∑ σk = √N σ 2 = √N σ k=1 (5) TLFeBOOK 34 Chapter 2 Mean Standard Deviation = 1 N N ∑σ k=1 k = 1 σ √N σ = N √N... Figure 2. 1A shows the distribution of 20 ,000 uniformly distributed random numbers between −1 and +1 The distribution is approximately flat between the limits of ±1 as expected When the data set consists of 20 ,000 numbers, each of which is the average of two uniformly distributed random numbers, the distribution is much closer to Gaussian (Figure 2. 1B, upper right) The distribution *Both terms are used and. .. constructed from 20 ,000 numbers that are averages of only 8 random numbers appears close to Gaussian, Figure 2. 1D, even though the numbers being averaged have a uniform distribution The probability of a Gaussianly distributed variable, x, is specified in the well-known normal or Gaussian distribution equation: p(x) = 1 σ 2 2 e−x /2 2 (1) TLFeBOOK Basic Concepts 33 Two important properties of a random variable... Figure 2. 3), this multiplication and summation is the same as scalar product of the two vectors.* The scalar product is defined as: ͫ ͬͫ ͬ a1 a2 Scalar product of a & b ≡ 〈a,b〉 = Ӈ an b1 b2 Ӈ bn = a1b1 + a2b2 + + anbn ( 12) Note that the scalar product results in a single number (i.e., a scalar), not a vector The scalar product can also be defined in terms of the magnitude of the two vectors and the . quantization step size, δ=V MAX /2 N . Assuming that the error is uniformly distributed between −δ /2 +δ /2, the variance, σ,is: σ= ∫ δ /2 −δ /2 η 2 /δ dη=V 2 Max (2 −2N )/ 12 (8) Assuming a uniform distribution,. box and record the following output: Frequency (Hz): 2 10 20 60 100 125 150 20 0 300 400 V out volts rms: .15×10 −7 0.1×10 −3 0.0 02 0 .2 1.5 3 .28 4.47 4.97 4.99 5.0 What is the cutoff frequency and. signal processing algorithms and data that can be used to evaluate a signal processing system under development. The Web is becoming an evermore useful tool in signal and image processing, and a