©) studocu Sampling & reconstruction‘Sampling and Reconstruction of Analog Signals With Aliasing CHAPTER 1 1, INTRODUCTION: A signal is defined as any physical quantity that varies with time, space, or any other independent variable or variables. Signals are classified into two types periodic signals and aperiodic signals. Periodic signals are defined as signals which repeat at time T. Aperiodic signals are defined as which don’t repeat at certain intervals of time. These signals are again classified into analog and digital signals, The continuous time signal is an analog and discrete time signal is a digital signal, The signals are functions of a continuous variable, such as time or space, and usually take on values in a continuous range. Such signals may be processed directly by appropriate analog systems such as filters or frequency analyzers or frequency multipliers for the purpose of changing their characteristics or extracting some desired information, Digital signal processing provides an alternative method for processing the analog signal. slog ea | Soa Apalog AD pie Dia Annee inapot convener f = “| converter ~ ees Se! poco | tel i / Digiel Digital inp cont om sien Fig.1.1 Block Diagram of digital signal processing ‘An analog signal is converted into a digital signal in A/D convertor by the following steps: 1. Sampling 2. Quantising 3. coding ‘The sampler samples the input signal with a sampling interval T. The output signal is discrete-in-time but continuous in amplitude .The output of the sampler is applied to the quantizer It converts the signal into Downloaded by Mishun M (mmishun761@gmail com) “ =‘Sampling and Reconstruction of Analog Signals With Aliasing discrete -time, discrete-amplitude signal. The final step is coding the coder maps each quantized sample value in digital word, Analog Dinceioume uamines Fig.1.2 Block diagram of analog to digital conversion 1, Sampling: This is the conversion of a continuous-time signal into a discrete time signal obtained by Ac time instants, taking “samples” of the continuous time signal at dis ‘Thus, if xa(t) is the input to the sampler, the output is xa (nT) = x(n), w here T is called the sampling interval. 2. Quantization: This is the conversion 0 f a discrete-time continuous-valued signal in to a discretetime, discrete-valued signal. The value of each signal sample is represented by a value selected from a finite set of possible values. The d difference between the un quantized sample x (n) an d the quantized output x q(n) is called the quantization error 3. Aliasing is an effect that causes different signals to become indistinguishable (or aliases of one another) when sampled., It also refers to the distortion that results when the signal reconstructed from samples is different from the original continuous signal (or) Aliasing is the generation of a false (alias) frequency along with the correct one when doing frequency sampling. Downloaded by Mishun M (mmishun781@gmail com), eee‘Sampling and Reconstruction of Analog Signals With Aliasing CHAPTER 2 2, SAMPLING: In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave (a continuous signal) to a sequence of samples (a discrete-time signal). A sample refers to a value or set of values at @ point in time and/or space. A sampler is a subsystem or operation that extracts samples from a continuous signal. A theoretical ideal sampler produces samples equivalent to the instantaneous value of the continuous signal at the desired points. X¢a) Xa(nT). -oOm is uniquely determined from its samples x(n'T),if the sampling frequency fs22fmaxi.e., sampling frequency must be at least twice the highest Downloaded by Mishun M (mmishun761@gmail com) “ =‘Sampling and Reconstruction of Analog Signals With Aliasing frequency present in the signal. Where fmax is the largest frequency component in the analog signal. With the sampling rate selected in this manner, any frequency component, say [Fil < fmax, in the analog signal is mapped into a discrete-time sinusoid with a frequency. Fmma is/2=1/2T Qmax-aFs-wT “12s Fi/FS<1/2 of -x 2fmax = 2B then Xa(t) can be exactly recovered from its sample values using the interpolation function. sinanat (= 2nBt Thus Xa(t) may be expressed as roan () Xag(t-~) When sampling of Xa(t) is performed at the minimum sampling rate Fs=2B then the reconstruction formula becomes n n FplenBee-) 0-2 GS sin2nB(t — X(S) —B BOF Fig. 3.1 Fourier Transform of a band limited function ‘The sampling rate Fn: =2fimax is called the Nyquist rate, The Nyquist rate, named after Harry Nyquist, is twice the bandwidth of a bandlimited function or a bandlimited channel, This term means two different things under two different circumstances: Downloaded by Mishun M (mmishun781@gmail com), meee‘Sampling and Reconstruction of Analog Signals With Aliasing 1, Asa lower bound for the sample rate for alias-ftee signal sampling” (not to be confused with the Nyquist frequency, which is half the sampling rate of a discrete-time system) and 2. As an upper bound for the symbol rate across a bandwidth-limited baseband channel such as a telegraph line or passbandchannel such as a limited radio frequency band or a frequency division multiplex channel, ‘The Bandwidth is aslo known as the Nyquist frequency and the twice the band width is known as the Nyquist rate. The sampling frequency must be exceeded in order to avoid the aliasing effects. CHAPTER 4 4, RECONSTRUCTION OF SIGNALS: We have discussed that a band limited signal x(t) can be reconstructed _ from its samples if the sampling rate is nyquist rate. This reconstruction is accomplished by passing the sampled signal through an ideal low pass filter of bandwidth D Hz, That sampled signal must be passed through an ideal low pass filter having bandwidth D Hz. and gain T. This is the description for the process of reconstruction in the frequency domain to find the DTFT of the discrete-time signal. Avsitinans | <8. Seeker | f#——* Anadeviena! SB. [ae] DSP SE} ae = suey | Posttar Recoveresenalag signal | dre paes) Fig.4.1 Reconstruction n the frequency domain to find DIFT This reconstruction can be thought of as a 2-step process + First the samples are converted into a weighted impulse train. yY x(n)8(t— as) = + x(-1)6(a + Ts) Downloaded by Mishun M (mmishun761@gmail com) “ =‘Sampling and Reconstruction of Analog Signals With Aliasing + Then the impulse train is filtered through an ideal analog lowpass filter band-limited to [- Fs/2, Fs/2] band. This two-step procedure can be described mathematically using an interpolating formula a(t) YX x(n)sine|Fs(t — n7'5)| As per the impact of phase on reconstruction of signals, 1, Phase of a signal helps us to achieve a desired signal-noise ratio. 2, Phase plays role in the amplitude estimation stage of single-channel speech enhancement and separation, 3. Replacing the noisy signal phase with an estimated phase can Iead to considerable improvement in the perceived signal quality, ‘There are many techniques that can be used to reconstruct a signal and the selection of the technique to be used is depends on what accuracy of reconstruction is required and how oversampled the signal is. Probably the simplest approximate reconstruction idea is to simply let the reconstruction always be the value of the ‘most recent sample. Its a simple technique because the samples in the form of numerical codes, can be the input signal to a Digital to Analog converter, which is clocked to produce a new output signal with every clock pulse. This technique produces a signal which has a stair step shape that follows the original signal. This type of signal reconstruction can be modeled except for quantization effects by passing the impulse sampled signal through a system called a zero order hold. The zero order hold causes a delay to the original signal because it is causal, Another natural reconstruction idea is to interpolate between samples with straight lines. This is obviously a better approximation of the original signal but it is a little harder to implement. This interpolation can be accomplished by following the zero orders hold by an identical zero order hold. This ‘means that the impulse response of such a signal reconstruction filter would be the convolution of the zero order hold impulse response with itself 4.1 Aliasing: In reconstructing a signal from its samples, there is another practical difficulty. The sampling theorem was proved on the assumption that the signal x(t) is bandlimited. All practical signals are time limited, i.., they are of finite duration. As a signal cannot be timelimited and bandlimited simultaneously. Thus, if a signal is timelimited, it cannot be bandlimited and vice versa (but it can be simultaneously non timelimited and non bandlimited), Clearly it can be said that all practical signals which are necessaily timelimited, are non Downloaded by Mishun M (mmishun781@gmail com), eee‘Sampling and Reconstruction of Analog Signals With Aliasing bandlimited, they have infinite bandwidth and the spectrum X(f) consists of overlapping cycles of X() repeating every f, Hz (sampling frequency). Because of infinite bandwidth, the spectral overlap will always be present regardless of what ever may be the sampling rate chosen. Because of the overlapping tails,X() has not complete information about X() and it is not possible, even theoretically to recover x(t) from the sampled signal x'(b. ‘The loss of the tail of X() beyond |f | > f/2. Hz. The reappearance of this tail inverted or folded onto the spectrum. The spectra cross at frequency f/2 = /2T Hz. This frequency is called the folding frequency. ‘The spectrum folds onto itself at the folding frequency. For instance, a component of frequency ({,/2)+ fx shows up as or act like a component of lower frequency (fy/2)- fx in the reconstructed signal. Thus the components of frequencies above f/2 reappear as components of frequencies below f,/2. This tail inversion is known as spectral folding or aliasing which is shown in Fig. 5. In this process of aliasing not only we are losing all the components of frequencies above f./2Hz, but these very components reappear as lower frequency components. This reappearance destroys the integrity of the lower frequency components. 4.2 Sampling and reconstruction in digit " digital signal spt] CP converter | — Weeser DC converter vi ult) Fig.4.2 ideal digital processing of analog signal 1. CD converter produces a sequence x[n] from x(\). 2. X{n] is processed in discrete-time domain to give y(n). u(t) = > ylkjsine 3, DC converted creates y(t) from yln). anti-aliasing analog-to-digital digital signal Adigital-to-analog Ty) iter [aq] converter [F") processor [ij"] converter Ti) TT ONTERSAT ELECIRONIES AND COMMUNICATION ENGINEERING Downloaded by Mishun M (mmishun761@gmail com)‘Sampling and Reconstruction of Analog Signals With Aliasing Fig.4.3 Practical digital processing of an analog signal Anti-aliasing filter is a filter which is used before a signal sampler, to restrict the bandwidth of a signal to approximately satisfy the sampling theorem. The potential defectors are all the frequency components beyond fi/2 Hz, We should have to eliminate these components from x(t) before sampling x(1). As a result of this we lose only the components beyond the folding frequency /2 Hz. These frequency components cannot reappear to corrupt the components with frequencies below the folding frequency. This suppression of higher frequencies can be accomplished by an ideal filter of bandwidths /72 Hz. This filter is called the anti-aliasing filter. The anti-aliasing operation must be performed before the signal is sampled, The anti- aliasing filter, being an ideal filter is unrealizable. In practice, we use a steep cutoff filter, which leaves a sharply attenuated residual spectrum beyond the folding frequency f¥/2. 1. X(t) may not be precisely band limited , a low pass filter or anti-aliasing filter is needed to process xi). Ideal CD converter is approximated by AD converter + Not practical to generate 3(t) + AD converter introduces quantization error. Ideal DC converter is approximated by DA converter because ideal reconstruction of is impossible 1 Not practical to perform infinite summation Not practical to have future data, Downloaded by Mishun M (mmichun781@gmail com), eens