EE 422G - Signals and Systems LaboratoryLab 1 Sampling and QuantizationWritten byKevin D. DonohueDepartment of Electrical and Computer EngineeringUniversity of KentuckyLexington, KY 40506January 17, 2013Objectives:Introduction to Matlab and Simulink features for signal analysis.1. Back groundSampling and Aliasing:This impulse train is illustrated in Fig. 2.Quantization and Uniformly Distributed Additive Noise.Quantizationts(t) -3T -2T -T 0 T 2T 3T ……1EE 422G - Signals and Systems LaboratoryLab 1 Sampling and QuantizationWritten byKevin D. DonohueDepartment of Electrical and Computer EngineeringUniversity of KentuckyLexington, KY 40506January 17, 2013Objectives:- Introduction to Matlab and Simulink features for signal analysis. - Apply Matlab to examine relationship between quantization bits and signal-to-noise ratios (SNR).- Apply Simulink to examine aliasing and the impact of non-ideal low-pass filtering forreconstructing a signal from its samples. 1. Back groundThis laboratory exercise focuses on the relationship between noise and interference resulting from digitizing an analog signal. Digitization results in 2 types of noise. The first is from sampling a continuous-time (CT) signal at discrete points in time, which causes aliasing. If the signal is not sampled at a rate higher than twice its highest frequency, then interference from aliasing may occur. The second type of noise results from rounding-off the sample amplitudes to discrete levels. This rounding error results in an additive noise process referred to as quantization noise.Sampling and Aliasing:A mathematical representation of the sampling operation is given by a sequence of impulse functions in continuous time:s(t )=∑n=−∞∞δ (t−nT ), (1)where the impulses are separated by T seconds (sampling interval) corresponding to a sampling frequency of Fs=1T Hz. The sampling function s(t) is shown in Fig. 1.Figure 1. Impulse train to model sampling operation.)(ˆfS -3Fs -2Fs -Fs 0 Fs 2Fs 3Fs ……fT1FNf-FN1Xc(f)It can be shown that the Fourier Transform of s(t) is also an impulse train in the frequency domain with impulses separated by the sampling frequency:^S (f )=1T∑k =−∞∞δ( f −kFs)(2)This impulse train is illustrated in Fig. 2.Figure 2. Fourier transform of sampling impulse train that creates aliased spectra.Now consider a CT signal, xc(t), that is bandlimited to FN:Xc(f)=0 for |f|>FN(3)where FNis referred to as the bandlimit. As an example, consider the Fourier Transform of bandlimited signal presented in Fig. 3. Figure 3. Example of Fourier Transform of a bandlimited continuous-time signal.Consider sampling xc(t) by multiplying it with the impulse train of Eq. (1), which effectively zeros out the information between sampling points:xs(t )=xc(t )∑n=−∞∞δ (t −nT ). (4)Now for a linear time-invariant (LTI) system, multiplication in the time domain corresponds to convolution in the frequency domain, given by:^Xs(f )=^Xc( f )∗1T∑k=−∞∞δ( f −kFs)=1T∫−∞∞^Xc( f −λ )∑k=−∞∞δ( λ−kFs)dλ. (5)FNf-FNXs(f)FS-2FS - FS2 FS……Xs(f)FS-FNfFS-2 FS-FS2 FS……3 FS-3FSSince the convolution of a signal with a shifted delta Dirac function, results in a shift version of the signal, Equation (5) shows that sampling replicates the original signal spectrum along the frequency axis separated by integer multiples of the sampling frequency Fs. The convolution integral of Eq. (5) becomes:^Xs(f )=1T∑k =−∞∞^Xc( f −kFs)(6)The aliasing described in Eq. (6) is illustrated in Fig. 4 for two cases. Figure 4a shows the case when the sampling frequency is greater than twiceFN. Figure 4b shows the case whenthe sampling frequency is less than twiceFN.The Nyquist frequency defined as half of the sampling frequency of a digital processing system. This is also referred to as the folding frequency since frequencies beyond this value fold back onto the non-aliased spectral range. Note that when the conditions on the samplingtheorem are met (i.e. Nyquist frequency is greater than FN), the aliased spectra do not interfere with each other and the original signal can be recovered from its samples with an idea low-pass filter. When overlap between aliased spectra occurs, the interference results inan irreversible noise/distortion process and the original signal cannot be recovered. Aliasing in this case irreversibly degrades the signal. Bandlimit FN is a property of the continuous-time signal and is sometimes referred to as the Nyquist rate. Note: The Nyquist frequency is aproperty of the processing system while Nyquist rate is a property of the continuous-time signal.(a)(b)Figure 4. Aliased spectra when sampling rate is (a) greater than twice the bandlimit;Fs>2 FN, (b) less than twice the bandlimit,Fs<2 FN.The derivation leading to Fig. 4 is the basis for the sampling theorem, which states that a bandlimited signal can be recovered from it samples if it is sampled at a rate greater than twice its highest frequency. The recovery can be performed with ideal low-pass filtering or sinc function interpolation. Note in Fig. 4a an ideal low-pass filter can be applied with a cut-off anywhere between FN and FS-FN. This way the original spectrum is undistorted and the aliased spectra are all eliminated. For the sampled signal of Fig. 4b, this is not possible because the original spectrum overlaps with the aliased spectra. No cutoff exists to eliminate the aliased spectra without distorting the original spectrum. Quantization and Uniformly Distributed Additive Noise.A discrete-time signal cannot be stored on a computer, since it has continuous amplitudes. Therefore, the amplitudes must also be converted to discrete levels to be stored as binary words in a computer memory. Figure 5 illustrates the quantization process.(a)(b)(c)Figure 5. (a) CT signal, (b) corresponding discrete-time signal, (c) digital signal quantized to 4 levels corresponding to a 2 bit word per sample.For or real signals the error from round-off to the nearest quantization level is effectively a random noise process. The error can be modeled by subtracting the distance from the discrete-time signal amplitude to the nearest quantization level, given by:nq(nT )=xc(nT )−xq(nT ), (7)where xq(nT) is the signal value after rounding off to a quantization level.Noise