This set of Signals & Systems Multiple Choice Questions & Answers (MCQs) focuses on “Dirichlet’s Conditions”.

1. How many dirichlet’s conditions are there?

a) One

b) Two

c) Three

d) Four

View Answer

Explanation: There are three dirichlet’s conditions. These conditions are certain conditions that a signal must possess for its fourier series to converge at all points where the signal is continuous.

2. What are the Dirichlet’s conditions?

a) Conditions required for fourier series to diverge

b) Conditions required for fourier series to converge if continuous

c) Conditions required for fourier series to converge

d) Conditions required for fourier series to diverge if continuous

View Answer

Explanation: Dirichlet’s conditions are Conditions required for fourier series to converge. That is there are certain conditions that a signal must posses for its fourier series to converge at all points where the signal is continuous.

3. What is the first Dirichlet’s condition?

a) Over any period, signal x(t) must be integrable

b) Multiplication of the signals must be continuous

c) x(t) should be continuous only

d) A signal can be integrable except break points

View Answer

Explanation: In the case of Dirichlet’s conditions, the first property leads to the integration of signal. It states that over any period, signal x(t) must be integrable.

That is ∫|x(t)|dt<∞.

4. Is dirichlet’s condition possible in case of discrete signals?

a) True

b) False

View Answer

Explanation: Dirichlet’s conditions is not possible in case of discrete signals. That is these are certain conditions that a signal must posses for its fourier series to converge at all points where the signal is continuous only.

5. What guarantees that coefficient is finite in a dirichlet’s condition?

a) First condition

b) Second condition

c) Third condition

d) Fourth condition

View Answer

Explanation: The first property is:

That is ∫|x(t)|dt<∞

Now, X

_{n}= 1/T∫|x(t)e

^{-jwt}|dt ≤ 1/T∫|x(t)|dt

So, X

_{n}<∞.

6. What is the second dirichlet’s condition?

a) In any finite interval, x(t) is of bounded variation

b) In most of a finite interval, x(t) is of bounded variation

c) In any finite interval, x(t) is of unbounded variation

d) In majority finite interval, x(t) is of unbounded variation

View Answer

Explanation: In any finite interval, x(t) is of bounded variation. That is there are no more than a finite number of maxima and minima during a single period of the signal.

7. There are maxima and minima not possible in dirichlet’s conditions.

a) True

b) False

View Answer

Explanation: Maxima and minima are possible if they are infinite number as stated by the second dirichlet’s condition. In any finite interval, x(t) is of bounded variation. That is there are no more than a finite number of maxima and minima during a single period of the signal.

8. What is the third dirichlet’s condition?

a) Finite discontinuities in the infinite interval

b) Finite discontinuities in the finite interval

c) Infiinite discontinuities in the infinite interval

d) Finite discontinuities in the all the intervals

View Answer

Explanation: The third condition states that in any finite interval of time, there is an only a finite number of discontinuities. Hence, finite discontinuities in the finite interval are the correct option.

9. In the third condition, does each of the discontinuities need to be finite?

a) All the time

b) Sometimes

c) Never

d) Rarely

View Answer

Explanation: The third condition states that in any finite interval of time, there is the only finite number of discontinuities. And furthermore, each of these discontinuities must be finite too.

10. What is the sum of the series at a point when the signal is discontinuous?

a) Average of the previous limits

b) Previous limits are considered

c) Future limits are added

d) Average of left hand limit and right hand limit are taken

View Answer

Explanation: If the signal is discontinuous at a point t, then an average of left hand limit and right hand limit of the signal x(t) are taken.

That is x(t) = ½[ x(t+)+x(t-)].

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