**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.

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

**Task1:**Consider an analog signal

*xa*(

*t*) = cos(20p

*t*+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 D

*t=*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.

**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

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