IMPACT OF PHASE ON SAMPLING AND RECONSTRUCTION OF SIGNALS

  ABSTRACT

Objectives:

1.      Generate discrete-time sequences from analog signals for various phase angles.

2.      Determine analog signals from discrete-time sequences using various interpolation filters.

3.      Study the effect of phase on reconstruction signals.

4.      Design a filter to remove noise

The tasks that are to be performed in this project are:

Task1: Consider an analog signal xa(t) = cos(20pt +f), 0 ≤t ≤1..Let  f= 0,p / 6,p / 4,

p / 3 and p / 2

Task2: This analog signal is sampled at Ts = 0.05 sec intervals to obtain x[n] Compute

x[n] from xa (t) for all the phase values. Plot x[n] and their spectrum.

Task3: Reconstruct the analog signal ya (t) from the samples x[n] using (a) Sync

(b) Cubic Spline interpolation filters. Use Dt= 0.001 sec.

Task4: Observe the resultant construction in each case that has the correct frequency

but a different amplitude. Explain these observations. Comment on the role of phase of

xa (t) on the sampling and reconstruction of signals.

Task5: Consider a AWGN is corrupted the signal with variance of 20, 30 dB. Plot the

noisy signal and its spectrum. Design a filter to remove the noise. Repeat the above

steps for each case.

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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.

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 discrete –time, discrete-amplitude signal. The final step is coding the coder maps each quantized sample value in digital word.

1.      Sampling: This is the conversion of a continuous-time signal into a discrete time signal

obtained by taking “samples’" of the continuous time signal at discrete time instants.

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 o f a discrete-time continuous-valued signal in to a discrete-time, 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.      Coding:  In the coding process, each discrete value x q{n) is represented by a 6-bit binary sequence.

 In this we explain about the sampling and different types of sampling and how to reconstruct a signal and the impact of phase on the sampling and reconstruction of signal

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 a 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(n) = Xa (nT).  -∞ < n < ∞

Where X( n ) is the discrete-time signal obtained by “ taking samples” of the analog signal Xa(t) every T seconds. The time interval T between successive samples is called the sampling period or sample interval and its reciprocal 1 / T = Fs is called the sampling rate.

T=nT=n/Fs

consider an analog sinusoidal signal of the form

Xa(t) = A cos ( 2π Ft + φ )

When sampled periodically at a rate Fs=1 / T samples per second

Xa( nT ) =x(n) = A cos(2πFnT +φ)=A cos (2πnF/Fs+φ)

By comparing both the Eq(1) and Eq(2) we obtain the relation between the frequency variables.

f=F/Fs

(or) ω=ΩT

             

where,

                   Fs=sampling frequency

                   F=frequency of analog

                   f=frequency of digital signal