Signals and Vectors
1. Define Signal.
A.
Signal is a physical quantity that varies with respect to time , space or any
other independent variable.
Or
It
is a mathematical representation of the system. Eg y(t) = t. and x(t)= sin t.
2. Define system?
A.
A set of components that are connected together to perform the particular task.
3. What are the major classifications of the signal?
A.
(i) Discrete time signal
(ii) Continuous time signal
4. Define discrete time signals and classify them.
A.
Discrete time signals are defined only at discrete times, and for these
signals, the
independent
variable takes on only a discrete set of values.
Classification
of discrete time signal:
1.Periodic
and Aperiodic signal
2.Even
and Odd signal
5. Define continuous time signals and classify them.
A.
Continuous time signals are defined for a continuous of values of the
independent variable. In the case of continuous time signals the independent
variable is continuous.
For
example:
(i)
A speech signal as a function of time
(ii)
Atmospheric pressure as a function of altitude
Classification
of continuous time signal:
(i)
Periodic and Aperiodic signal
(ii)
Even and Odd signal
6. Define discrete time unit step &unit impulse.
A.
Discrete time Unit impulse is defined as δ[n]= {0, n≠ 0
{1,
n=0
Unit
impulse is also known as unit sample.
Discrete
time unit step signal is defined by
U[n]= {0,n=0
{1,n>=
0
7. Define continuous time unit step and unit
impulse.
A.
Continuous time unit impulse is defined as
δ(t)= {1, t=0
{0,
t ≠ 0
Continuous
time Unit step signal is defined as
U(t)= {0, t<0
{1,
t≥0
8. Define unit ramp signal.
A.
Continuous time unit ramp function is defined by
r(t)= {0,t<0
{t,
t≥0
A
ramp signal starts at t=0 and increases linearly with time ‘t’.
9. Define periodic signal. and nonperiodic signal.
A.
A signal is said to be periodic ,if it exhibits periodicity.i.e., X(t +T)=x(t),
for all values of t. Periodic signal has the property that it is unchanged by a
time shift of T. A signal that does not satisfy the above periodicity property
is called an aperiodic signal.
10. Define even and odd signal ?
A.
A discrete time signal is said to be even when, x[-n]=x[n].
The
continuous time signal is said to be even when, x(-t)=
x(t)
For example,Cosωn is an even signal.
The
discrete time signal is said to be odd when x[-n]=
-x[n]
The
continuous time signal is said to be odd when x(-t)=
-x(t)
Odd
signals are also known as nonsymmetrical signal.
Sine
wave signal is an odd signal.
11. Define Energy and power signal.
A.
A signal is said to be energy signal if it have finite energy and zero power.
A
signal is said to be power signal if it have infinite energy and finite power.
If
the above two conditions are not satisfied then the signal is said to be
neither energy nor power signal
12. Define unit pulse function.
A.
Unit pulse function Π(t)
is obtained from unit step signals
Π(t)=u(t+1/2)-
u(t-1/2)
The
signals u(t+1/2) and u(t-1/2) are the unit step signals shifted by 1/2units in
the time axis towards the left and right ,respectively.
13. Define continuous time complex exponential
signal.
A.
The continuous time complex exponential signal is of the form
x(t)=Ceat
where
c and a are complex numbers.
14. What is continuous time real exponential signal.
A.
Continuous time real exponential signal is defined by
x(t)=Ceat where c and a are complex numbers. If
c and a are real, then it is called as real exponential.
15. What is continuous time growing exponential
signal?
A.
Continuous time growing exponential signal is defined as
x(t)=Ceat where c and a are complex numbers. If
a is positive, as t increases, then x(t) is a growing exponential.
16. What is continuous time decaying exponential?
A.
Continuous time growing exponential signal is defined as
x(t)=Ceat where c and a are complex numbers. If
a is negative, as t increases, then x(t) is a decaying exponential.
17. Check Whether the given system is causal and
stable.
A. y ( n ) = 3 x ( n - 2) + 3 x ( n + 2 )
Since y (n) depends upon x(n+2), this
system is noncausal. As long as x (n – 2) and
x(n + 2) are bounded, the output y(n)
will be bounded. Hence this system is stable.
18. What is the periodicity of x(t) = ej100Πt + 30 ?
A. Here x(t) = ej100Πt + 30
Comparing above equation with ejωt+Ф, we get ω = 100 Π. Therefore period T is given as,
T=2 Π/ ω = 2 Π /100 Π = 1/50 = 0.02 sec.
19. Find the fundamental period of the signal x(n)=
3 ej3Π(n+1/2)
A. X(n) = 3/5 ej3Πn. ej3Π/2 = -j3/5 ej3Πn
Here, ω=3Π, hence, f=3/2=k/N. Thus the
fundamental period is N = 2.
20. Is the discrete time system describe by the
equation y (n) = x(-n) causal or non causal ?
Why?
A. Here y(n) = x(-n). If n = -2 then, y(-2
)= x(2). Thus the output depends upon future inputs. Hence system is noncausal.
21. Is the system describe by the equation y(t) =
x(2t) Time invariant or not? Why?
A. Output for delayed inputs
becomes,y(t,t1) =x(2t-t1). Delayed
output will be, y(t-t1)
= x[2(t-t1)]. Since y(t,t1) ≠ y(t-t1)
. The system is shift variant.
22.
What is analogy between vector and signal?
A
vector of the kind used in geometrical applications can be used to indicate a
time-varying sinusoidal signal or a fixed two dimensional image or a three
dimensional point. But when a signal, such as arising from a vibration or a
seismic signal or even a voice signal, is to be represented.
The
analogy between the sets of points of the signal and the vector of points is
now clear. The purpose of visualizing such an analogy is to mathematically
perform such operations on the signal which would convert the signal
information in a more useful form. A filter which operates on the signal does
it. A Fourier transform which operates on the vector of data tells the
frequencies present in the signal.
23.
Give an example to show analogy between vector and signal
A.
Nuclear magnetic resonance of a chemical, dichloro-ethane from such a
spectrometer instrument.
24.
What are ‘Orthogonal’ functions?
A.
The sine and cosine functions belong to this category. We can therefore
represent a signal as a sum of a number of sine and cosine waves sin θ, sin 2θ,
sin 3θ... cos θ, cos 2θ, cos 3θ... etc. Here θ denotes the time variable, by
the usual θ = ωt relationship.
25.
Advantage of orthogonality?
A.
Advantage of orthogonal functions is the coefficients are independent and so,
if a fit has been made with an mth-degree polynomial in P, and it is decided
later to use a higher degree, giving more terms, only the additional
coefficients are required to be calculated and those already calculated remain
unchanged.
26. What is purpose of Fourier series?
A. Used to analyze periodic signals, harmonic
constant of signals is analyzed and it can be developed for continuous time as
well as discrete time signals.
27. What are the types of Fourier series?
A.
1. Exponential Fourier series
2. Trigonometric Fourier series
28.Write down the exponential form of the Fourier
series representation of a periodic signal?
A. The equation is given by,
Here
the summation is taken from -
to
Here
the integration is taken from 0 to T. The set of coefficients { X(k)} are often
called the Fourier series coefficients or spectral coefficients. The
coefficient ao is
the dc or constant component of x(t).
29.Write short notes on dirichlets conditions for
fourier series.
A.
a. x(t) must be absolutely integrable
b.
The function x(t) should be single valued within the interval T.
c.
The function x(t) should have finite number of discontinuities in any
finite
interval of time T.
d.
The function x(t) should have finite number of maxima &minima in the
interval
T.
30.
What do odd functions have in Fourier series?
A.
It has only Sine terms. i.e., b(k)
33.
Which terms will be left for a even function
in Fourier series?
A.
It consists of only cosine terms. i.e., a(k)
34.
Explain half symmetry.
A.
Signal x(t) is said to be half symmetry if x(t)
= - x(t ± T/2)
a(0) = 0 and a(k) = b(k) = 0 for
even ‘k’
a(k)
=
for ‘k’ odd
b(k)
=
for ‘k’ even; a(k) and
b(k) contain only odd harmonics
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