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**Signals and Systems MCQs – Fourier Series MCQs ( Signals and Systems ) MCQs**

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# Periodic Signals – 1

**1. What are periodic signals?**

a) The signals which change with time

b) The signals which change with frequency**c) The signal that repeats itself in time**

d) The signals that repeat itself over a fixed frequency**Answer: c****Explanation: Those signals which repeat themselves in a fixed interval of time are called periodic signals. The continuous-time signal x(t) is periodic if and only if****x(t+T)= x(t).****2. Periodic signals are different in case of continuous time and discrete time signals.**

a) True**b) False****Answer: b****Explanation: Periodic signals are same in case of continuous time and discrete time signals.****In case of continuous time signal, x(t)=x(t+T), for all t>0****In case of discrete time signal,****x(n)=x(n+N), for all n>0.****3. What is the time period of a periodic signal in actual terms?****a) The signals which start at t=-∞ and end at t=+∞**

b) The signals which have a finite interval of occurrence

c) The signals which start at t= -∞ and ends at a finite time period

d) The signals which have a short period of occurrence**Answer: a****Explanation: The periodic signals have actually a time period between t=-∞ and at t= + ∞. These signals have an infinite time period, that is periodic signals are actually continued forever. But this is not possible in case of real time signals.****4. Periodic signals actually exist according to a definition.**

a) True**b) False****Answer: b****Explanation: Periodic signals are defined as signals having time period in between t=-∞ and t=+ ∞. These signals have an infinite time period that is periodic signals are continued forever. But real time signals always cease at some time due to distortion and resistance.****5. What is a fundamental period?**

a) Every interval of a periodic signal

b) Every interval of an aperiodic signal**c) The first interval of a periodic signal**

d) The last interval of a periodic signal**Answer: c****Explanation: The first time interval of a periodic signal after which it repeats itself is called a fundamental period. It should be noted that the fundamental period is the first positive value of frequency for which the signal repeats itself.****6. Comment on the periodicity of a constant signal?**

a) It is periodic**b) It is not periodic**

c) It is a mixture of period and aperiodic signal

d) It depends on the signal**Answer: b****Explanation: A constant signal is not periodic. It is because it does not repeat itself over in time. It is constant at any time, it is aperiodic.****7. A discrete time periodic signal is defined as x(n)= x(n+N)****How is the N defined here?****a) Samples/ cycle**

b) Samples/ twice cycle

c) Fundamental period

d) Rate of change of the period**Answer: a****Explanation: The value of N is a positive integer and it represents the period of any discrete time periodic signal measured in terms of number of sample spacing ( samples/cycle). The smallest value of N is a fundamental period.****8. What is the general range of a period of a signal?**

a) It can have of any value from positive to negative

b) It can be negative

c) It can be positive**d) It is always positive****Answer: d****Explanation: The period of a periodic signal is always positive. The smallest positive value of a periodic interval is called a fundamental period in case of both discrete and continuous time signal.****9. What is the area of a periodic signal in a periodic interval?**

a) It depends on the situation**b) It is same as the area in the previous interval**

c) It is different in different situations

d) It is the square of the fundamental period**Answer: b****Explanation: The area of any periodic signal in any interval is the same. Hence it is same as the previous interval. This results from the fact that a periodic signal takes same values at the intervals of T.****10. When is the sum of M periodic signals periodic?**

a) T/T_{i} = 1

b) T/T_{i} = 4**c) T/T _{i} = n_{i}**

d) T/T

_{i}= m+n

**Answer: c**

**Explanation: The sum of M periodic signal is not necessarily periodic. It is periodic only with the condition that**

**T/T**

_{i}= n_{i}, 1≤i≤M,**where T**

_{i}is the period of the signal and in the sum of n_{i}is an integer.**11. How is the period of the sum signal computed as?**

**a) T*n**

b) T*T

c) T*N+M

d) T *(n+m)

**Answer: a**

**Explanation: If a signal is periodic then we have to convert each of the periods to the ratio of integers. We have to take the ratio of greatest common divisor(gcd) from the numerator to the gcd of denominator. The LCM of the denominators of the resulting ratios is the value of n the period of the sum signal is T*n.**

**12. What is the necessary and sufficient condition for a sum of a periodic continuous time signal to be periodic?**

a) Ratio of period of the first signal to period of other signals should be constant

b) Ratio of period of the first signal to period of other signals should be finite

c) Ratio of period of the first signal to period of other signals should be real

**d) Ratio of period of first signal to period of other signal should be rational**

**Answer: d**

**Explanation: The necessary and sufficient condition for a sum of a periodic continuous time signal to be periodic is that the ratio of a period of the first signal to the period of other signals should be rational.**

**I.e T/T**

_{i}= a rational number.**13. Under what conditions the three signals x(t), y(t) and z(t) with period t1 t2 and t3 respectively are periodic?**

a) t1/t2= t2/t3

b) t1/t2 is rational

**c) All the ratios of the three periods in any order is rational**

d) t1/t2/t3= rational

**Answer: c**

**Explanation: if x(t) , y(t) and z(t) are to be periodic then,**

**t1/t2 should be rational and simultaneously**

**t1/t3 should be rational and**

**t2/t3 should be rational. Hence, all the ratios of the three periods in any order is rational.**

**14. What is the fundamental period of the signal : e**

^{jwt}?**a) 2π/w**

b) 2π/w2

c) 2π/w3

d) 4π/w

**Answer: a**

**Explanation: The complex exponential signal can be represented as**

**e**

^{jwt}= e^{jwt+jwT}**Hence, wt=2 π,**

**T= 2π/w.**

**15. What is the period of the signal :je**

^{jw11t}?a) 2π/10

**b) 2π/11**

c) 4π/10

d) 4π/11

**Answer: b**

**Explanation: From the definition of periodic signal, we express a periodic exponential signal as :**

**e**

^{jw11t}= e^{jwt+jwT}**Hence, 11wt=2 π,**

**which gives the fundamental period as**

**2π/11.**

# Periodic Signals – 2

**1. Is the sum of discrete time periodic signals periodic?**

a) No, they are not**b) Yes they are**

c) Depends on the signal

d) Not periodic if their ratio is not rational**Answer: b****Explanation: The sum of discrete time periodic signals always periodic because the period ratios N/N are always rational.****For the continuous time, it depends on the ratio.****2. How can we generate a periodic signal from a periodic signal itself?****a) By extending a signal with duration T**

b) Cannot be extended

c) By extending the periodic signal’s amplitude

d) By extending the sugar with duration 2π**Answer: a****Explanation: A periodic signal x(t) can be generated by a periodic extension of any segment of x of duration T( the period).****As a result, we can generate x(t) from any segment x(t) having a duration of one period by replacing this segment and reproduction thereof end to end ad infinitum on either side.****3. Is a non periodic signal same as aperiodic signal?**

a) No, it is not same as an aperiodic signal**b) Yes it is the other name of aperiodic signal**

c) It is a branch of aperiodic signal

d) Aperiodic signal is a branch of non periodic signal**Answer: b****Explanation: A signal which does not satisfy the condition:****x(t) = x(t+T) is called an aperiodic signal.****Non periodic is another name of an aperiodic signal. Hence it is exactly the same.****4. What is the period of the signal: 2cost/6?**

a) 8π

b) 16π**c) 12π**

d) 10π**Answer: c****Explanation: Comparing the above signal with the standard form Acos2πFt, where A is the amplitude and F is the frequency,****We get, 2πF=⅙****So, F= 1/12π****Hence, t= 12π.****5. When a continuous signal is a mixture of two continuous periodic signals, what is its periodicity?**

a) LCM of the periods of the two signals, provide their ratio is unity**b) LCM of the periods of the two signals, provide their ratio is rational**

c) HCF of the periods of the two signals, provide their ratio is rational

d) LCM of the periods of the two signals provide their ratio is real**Answer: b****Explanation: When a continuous signal is a mixture of two continuous periodic signals if their time periods are T1 and T2, and their ratio is rational number, then, the periodicity of the continuous time signal will be the LCM of T1 and T2.****6. Is the signal e ^{αt} periodic?**

a) Not periodic

b) Yes periodic

**c) Depends on the value of**

d) Semi- periodic

**Answer: c**

**Explanation: Using the definition of x(t),**

**x(t) = e**

^{αt}**e**

^{jwt}= e^{jwt+jwαT}**For any value of α, if alpha is positive, it has a remaining term e**

^{jwαT}**Hence it is not periodic.**

**7. What is a fundamental angular frequency?**

a) The inverse of the fundamental time period

b) The inverse of fundamental frequency

**c) Fundamental frequency in radians**

d) Fundamental frequency in degree

**Answer: c**

**Explanation: The inverse of the fundamental time period is called fundamental frequency. If it is F, then 2πF is called the fundamental angular frequency ie it is a fundamental frequency in radians.**

**8. What is the period of cos3t + sin14t?**

a) 4π

**b) π**

c) 2π

d) 3π

**Answer: b**

**Explanation: We know, T1= 2 π/3 and T2= 2 π/14**

**Now, T1/T2=14/3.**

**So, LCM gives the time period as π.**

**9. What is the periodicity of a discrete time signal?**

a) 2πm/w

**b) 2πm/w**

c) 2πm/w

d) 2πm/w

**Answer: b**

**Explanation: Using exponential function, we can show that**

**2π/N= w/m**

**Which when rearranged gets us 2πm/w.**

**10. What is the condition of a periodicity of exponential signal e ^{αt}?**

**a) α=1**

b) α=2

c) α=3

d) Depends on equation

**Answer: a**

**Explanation: From, x(t+T)= e**

^{αt+T}= e^{αt}e^{αt}. For any value of α, e^{αt}≠1 so x(t+T) ≠x(t). So only if α=1, the signal will be periodic.# Fourier Series

**1. What is Fourier series?**

a) The representation of periodic signals in a mathematical manner is called a Fourier series

b) The representation of non periodic signals in a mathematical manner is called a Fourier series

c) The representation of non periodic signals in terms of complex exponentials or sinusoids is called a Fourier series**d) The representation of periodic signals in terms of complex exponentials or sinusoids is called a Fourier series****Answer: d****Explanation: The Fourier series is the representation of non periodic signals in terms of complex exponentials, or equivalently in terms of sine and cosine waveform leads to Fourier series. In other words, Fourier series is a mathematical tool that allows representation of any periodic wave as a sum of harmonically related sinusoids.****2. Who discovered Fourier series?**

a) Jean Baptiste de Fourier**b) Jean Baptiste Joseph Fourier**

c) Fourier Joseph

d) Jean Fourier**Answer: b****Explanation: The Fourier series is the representation of non periodic signals in terms of complex exponentials or sine or cosine waveform. This was discovered by Jean Baptiste Joseph Fourier in 18th century.****3. Fourier series representation can be used in case of Non-periodic signals too. True or false?**

a) True**b) False****Answer: b****Explanation: False. The Fourier series is the representation of periodic signals in terms of complex exponentials, or equivalently in terms of sine and cosine waveform leads to Fourier series. In other words, Fourier series is a mathematical tool that allows representation of any periodic wave as a sum of harmonically related sinusoids. They are for periodic signals only.****4. What are the conditions called which are required for a signal to fulfil to be represented as Fourier series?****a) Dirichlet’s conditions**

b) Gibbs phenomenon

c) Fourier conditions

d) Fourier phenomenon**Answer: a****Explanation: When the Dirichlet’s conditions are satisfied, then only for a signal, the fourier series exist. Fourier series is of two types- trigonometric series and exponential series.****5. Choose the condition from below that is not a part of Dirichlet’s conditions?**

a) If it is continuous then there are a finite number of discontinuities in the period T1

b) It has a finite average value over the period T

c) It has a finite number of positive and negative maxima in the period T**d) It is a periodic signal****Answer: d****Explanation: Even if the Fourier series demands periodicity as the major necessity for its formation still it is not a part of Dirichlet’s condition. It is the basic necessity for Fourier series.****6. What are the two types of Fourier series?****a) Trigonometric and exponential**

b) Trigonometric and logarithmic

c) Exponential and logarithmic

d) Trigonometric only**Answer: a****Explanation: The two types of Fourier series are- Trigonometric and exponential. The exponential is more convenient for Fourier series calculations.****7. How is a trigonometric Fourier series represented?****a) A _{0} +∑[ancos(w_{0}t)+ ansin(w_{0}t)]**

b) ∑[ancos(w

_{0}t)+ ansin(w

_{0}t)]

c) A

_{0}*∑[ancos(w

_{0}t)+ ansin(w

_{0}t)]

d) A

_{0}+∑[ancos(w

_{0}t)+ ansin(w

_{0}t)] + sinwt

**Answer: a**

**Explanation: A**

_{0}+ ∑[ancos(w_{0}t)+ ansin(w_{0}t)] is the correct representation of a trigonometric Fourier series. Here A_{0}= 1/T∫x(t)dt and an =2/T∫x(t)cos(w_{0}t)dt and bn= 2/T∫x(t)sin(w_{0}t)dt.**8. How is the exponential Fourier series represented?**

a) X(t) = ∑X

_{n}e

^{jnwt}+ wt

b) X(t) = 1/T∑Xne

^{jnwt}

**c) X(t) = ∑X**

_{n}e^{jnwt}d) X(t) = T*∑X

_{n}e

^{jnwt}

**Answer: c**

**Explanation: The exponential Fourier series is represented as – X(t)=∑X**

_{n}e^{jnwt}. Here, the X(t) is the signal and X_{n}=1/T∫x(t)e^{-jnwt}.**9. What is the equation – X(t)=∑X**

_{n}e^{jnwt}called?**a) Synthesis equation**

b) Analysis equation

c) Frequency domain equation

d) Discrete equation

**Answer: a**

**Explanation: The equation – X(t) = ∑X**

_{n}e^{jnwt}called the synthesis equation of an exponential Fourier series. It is because it is used to synthesize the Fourier series.**10. What is the equation – X**

_{n}=1/T∫x(t) e^{jwtn}called?a) Synthesis equation

**b) Analysis equation**

c) Frequency domain equation

d) Discrete equation

**Answer: b**

**Explanation: The equation – X**

_{n}=1/T∫x(t)e^{-jwtn}called the analysis equation of an exponential Fourier series. It is because it is used to synthesize the Fourier series.# Fourier Series & Coefficients

**1. What are fourier coefficients?**

a) The terms that are present in a fourier series

b) The terms that are obtained through fourier series**c) The terms which consist of the fourier series along with their sine or cosine values**

d) The terms which are of resemblance to fourier transform in a fourier series are called fourier series coefficients**Answer: c****Explanation: The terms which consist of the fourier series along with their sine or cosine values are called fourier coefficients. Fourier coefficients are present in both exponential and trigonometric fourier series.****2. Which are the fourier coefficients in the following?****a) a _{0}, a_{n} and b_{n}**

b) a

_{n}

c) b

_{n}

d) a

_{n}and b

_{n}

**Answer: a**

**Explanation: These are the fourier coefficients in a trigonometric fourier series.**

**a**

_{0}= 1/T∫x(t)dt**a**

_{n}= 2/T∫x(t)cos(nwt)dt**b**

_{n}= 2/T∫x(t)sin(nwt)dt**3. Do exponential fourier series also have fourier coefficients to be evaluated.**

**a) True**

b) False

**Answer: a**

**Explanation: The fourier coefficient is : X**

_{n}= 1/T∫x(t)e^{-njwt}dt.**4. The fourier series coefficients of the signal are carried from –T/2 to T/2.**

**a) True**

b) False

**Answer: a**

**Explanation: Yes, the coefficients evaluation can be done from –T/2 to T/2. It is done for the simplification of the signal.**

**5. What is the polar form of the fourier series?**

**a) x(t) = c**

_{0}+ ∑cncos(nwt+ϕ_{n})b) x(t) = c

_{0}+ ∑cncos(ϕ

_{n})

c) x(t) = ∑cncos(nwt+ϕ

_{n})

d) x(t) = c

_{0}+ ∑cos(nwt+ϕ

_{n})

**Answer: a**

**Explanation: x(t) = c**

_{0}+ ∑cncos(nwt+ϕ_{n}), is the polar form of the fourier series.**C**

_{0}=a_{0}and c_{n}= √a2_{n}+ b2_{n}for n≥1**And ϕ**

_{n}= tan^{-1}b_{n}/a_{n}.**6. What is a line spectrum?**

a) Plot showing magnitudes of waveforms are called line spectrum

**b) Plot showing each of harmonic amplitudes in the wave is called line spectrum**

c) Plot showing each of harmonic amplitudes in the wave is called line spectrum

d) Plot showing each of harmonic amplitudes called line spectrum

**Answer: b**

**Explanation: The plot showing each of harmonic amplitudes in the wave is called line spectrum. The line rapidly decreases for waves with rapidly convergent series.**

**7. Fourier series is not true in case of discrete time signals.**

a) True

**b) False**

**Answer: b**

**Explanation: Fourier series is also true in case of discrete time signals. They just need to follow the dirichlet’s conditions.**

**8. What is the disadvantage of exponential Fourier series?**

a) It is tough to calculate

b) It is not easily visualized

**c) It cannot be easily visualized as sinusoids**

d) It is hard for manipulation

**Answer: c**

**Explanation: The major disadvantage of exponential Fourier series is that it cannot be easily visualized as sinusoids. Moreover, it is easier to calculate and easy for manipulation leave aside the disadvantage.**

**9. Fourier series uses which domain representation of signals?**

a) Time domain representation

**b) Frequency domain representation**

c) Both combined

d) Neither depends on the situation

**Answer: b**

**Explanation: Fourier series uses frequency domain representation of signals. X(t)=1/T∑X**

_{n}e^{jnwt}. Here, the X(t) is the signal and X_{n}= 1/T∫x(t)e^{-jwtn}.**10. How does Fourier series make it easier to represent periodic signals?**

**a) Harmonically related**

b) Periodically related

c) Sinusoidally related

d) Exponentially related

**Answer: a**

**Explanation: Fourier series makes it easier to represent periodic signals as it is a mathematical tool that allows the representation of any periodic signals as the sum of harmonically related sinusoids.**

# Fourier Series Coefficients – 2

**1. The Fourier series coefficient for the signal 10δ(t) is ___________**

a) 1

b) Cos (\frac{π}{2} k)

c) sin (\frac{π}{2} k)**d) 2****Answer: d****Explanation: X[k] = \frac{1}{T} \displaystyle\int_{-\frac{T}{2}}^{\frac{T}{2}} Aδ(t)e^{-jkωt} \,dt****= \frac{A}{2}****Here, A=10, T=5****∴ X[k] = 2.****2. The Fourier series coefficient for the periodic rectangular pulses of height 2A is ____________**

a) \frac{2A}{jkπ} \,sin\, \frac{π}{2} k

b) \frac{2A}{jkπ} \,cos\, \frac{π}{2} k**c) \frac{2A}{kπ} \,sin\, \frac{π}{2} k**

d) \frac{2A}{kπ} \,cos\, \frac{π}{2} k**Answer: c****Explanation: X[k] = \frac{1}{T} \displaystyle\int_{-\frac{T}{2}}^{\frac{T}{2}} x(t)e^{-jkωt} \,dt****= \frac{1}{T} \displaystyle\int_{-\frac{T}{4}}^{\frac{T}{4}} Ae^{-jkωt} \,dt****= \frac{2A}{T}[\frac{e^{-jkωt}}{-jkω}] (from –\frac{T}{4} to \frac{T}{4})****=\frac{2A}{kπ} \,sin\, \frac{π}{2} k.****3. The Fourier series coefficient for the periodic signal x(t) = sin ^{2}t is _____________**

**a) –\frac{1}{4} δ[k-1] + \frac{1}{2} δ[k] – \frac{1}{4} δ[k+1]**

b) –\frac{1}{4} δ[k-2] + \frac{1}{2} δ[k] – \frac{1}{4} δ[k+2]

c) –\frac{1}{2} δ[k-1] + δ[k] – \frac{1}{2} δ[k+1]

d) –\frac{1}{2} δ[k-2] + δ[k] – \frac{1}{2} δ[k+2]

**Answer: a**

**Explanation: sin**

^{2}t = (\frac{e^{jt} – e^{-jt}}{2j})^2**= –\frac{1}{4} (e**

^{2jt}– 2 + e^{-2jt})**The fundamental period of sin2t is π and ω = \frac{2π}{π} = 2**

**∴ X[k] = –\frac{1}{4} δ[k-1] + \frac{1}{2} δ[k] – \frac{1}{4} δ[k+1].**

**4. The Fourier series coefficient of time domain signal x (t) is X[k] = jδ[k-1] – jδ[k+1] + δ[k+3] + δ[k-3], the fundamental frequency of the signal is ω=2π. The signal is ___________**

a) 2(cos 3πt – sin πt)

b) -2(cos 3πt – sin πt)

**c) 2(cos 6πt – sin 2πt)**

d) -2(cos 6πt – sin 2πt)

**Answer: c**

**Explanation: x(t)=∑∞k=−∞X[k]ej2πkt**

**= je**

^{j2πt}– je^{-j2πt}+ e^{j6πt}+ e^{-j6πt}**= 2(cos 6πt – sin 2πt).**

Three Domain Representation For LTI Systems MCQs

**5. The Fourier series coefficient of time domain signal x (t) is X[k] = (−13)|k|. The fundamental frequency of signal is ω=1. The signal is _____________**

a) 45+3sin

b) 54+3sint

c) 54+3cost**d) 45+3sint****Answer: d****Explanation: x(t)=∑∞k=−∞X[k]ejkt****Or, x (t) = ∑−1k=−∞(−13)−kejk+∑∞k=0(−13)kejkt****= −13e−jt1+13e−jt+11+13ejt****= 45+3sint.****6. The Fourier series coefficient of the signal y(t) = x(t-t _{0}) + x(t+t_{0}) is _____________**

**a) 2 cos (2πt kt**

_{0}) X[k]b) 2 sin (2πt kt

_{0}) X[k]

c) 2 cos (2πt kt

_{0})

d) 2 sin (2πt kt

_{0})

**Answer: a**

**Explanation: x (t-t**

_{0}) is periodic with period T. the Fourier series coefficient of x (t-t_{0}) is X_{1}[k] = 1T ∫ x (t-t_{0})e^{-jkωt}dt**= e**

^{-jkωt0}X[k]**Similarly, the Fourier series coefficient of x (t+t**

_{0}) is X_{2}[k] = e^{jkωt0}X[k]**The Fourier series coefficient of x (t-t**

_{0}) + x (t+t_{0}) is**Y[k] = X**

_{1}[k] + X_{2}[k]**= e**

^{-jkωt0}X[k] + e^{jkωt0}X[k]**= 2 cos (2πt kt**

_{0}) X[k].**7. The Fourier series coefficient of the signal y(t) = Even{x(t)} is ___________**

**a) X[k]+X[−k]2**

b) X[k]−X[−k]2

c) X[k]+X∗[−k]2

d) X[k]−X∗[−k]2

**Answer: a**

**Explanation: even {x (t)} = \frac{x(t)+x(-t)}{2}**

**The Fourier series coefficient transform of x (t) is**

**X**

_{1}[k] = \frac{1}{T} ∫ x (-t)e^{-jkωt}dt**= \frac{1}{T} ∫ x (α)e**

^{jkωα}dα**= X [-k] ∴ The Fourier coefficient of Even{x(t)} = Y[k] = \frac{X[k]+X[-k]}{2}.**

**8. The Fourier series coefficient of the signal y(t) = Re{x(t)} is ____________**

a) X[k]+X[−k]2

b) X[k]−X[−k]2

**c) X[k]+X∗[−k]2**

d) X[k]−X∗[−k]2

**Answer: c**

**Explanation: Re{x (t)} = x(t)+x∗(−t)2**

**The Fourier coefficient of x* (t) is**

**X**

_{1}[k] = 1T ∫ x* (t)e^{-jkωt}dt = X∗1 [-k]**Or, X∗1 [k] = 1T ∫ x(t)e**

^{jkωt}dt = X [-k]**So, X**

_{1}[k] = X∗1 [-k]**∴ Y[k] = X[k]+X∗[−k]2.**

**9. The Fourier series coefficient of the signal y(t) = d2x(t)dt2 is _____________**

a) (2πkT)2X[k]

**b) –(2πkT)2X[k]**

c) j(2πkT)2X[k]

d) -j(2πkT)2X[k]

**Answer: b**

**Explanation: x(t)=∑∞k=−∞X[k]ej2πTkt**

**Now, dx(t)dt=−j(2πT)k∑∞k=−∞X[k]ej2πTkt**

**And, d2x(t)dt2=−(2πT)2k2∑∞k=−∞X[k]ej2πTkt**

**∴ Y[k] = – (2πkT)2X[k].**

**10. The Fourier series coefficient of the signal y(t) = x(4t-1) is ______________**

a) 8πTX[k]

b) 4πTX[k]

**c) e−jk8πTX[k]**

d) ejk8πTX[k]

**Answer: c**

**Explanation: The period of x (4t) is a fourth of the period of x (t). The Fourier series coefficient of x (4t) is still X[k]. Hence, the coefficient of x (4t-1) is e−jk8πTX[k].**

**11. The discrete time Fourier coefficients of ∑∞m=−∞δ[n−4m] is ____________**

a) –14 for all k

**b) 14 for all k**

c) –12 for all k

d) 12 for all k

**Answer: b**

**Explanation: N=4, ω = 2π4=π2**

**X[k]=14∑3n=4x[n]e−j(π2)nk**

**= 14 x[0] = 14 for all k.**

**12. The discrete time Fourier coefficient of cos**

^{2}(π8 n) is ______________a) π2(δ(k+1] + 2δ[k] + δ[k-1])

b) 14j(δ(k+1] + 2δ[k] + δ[k-1])

**c) 14(δ(k+1] + 2δ[k] + δ[k-1])**

d) π4(δ(k+1] + 2δ[k] + δ[k-1])

**Answer: c**

**Explanation: N=8, ω = 2π8=π4**

**X[n] = cos**

^{2}(π8 n) = 14(ej(π8)n+e−j(π8)n)2**=14(ej(π8)n+2+e−j(π8n)2**

**Or, X[k] = 14(δ(k+1] + 2δ[k] + δ[k-1]).**

**13. V(t) = 5, 0≤t<1;**

**t, t≥1;**

**The Laplace transform of V (t) is ___________**

a) 5s+e−ss2+4e−ss

**b) 5s+e−ss2–4e−ss**

c) 5s–e−ss2–4e−ss

d) 5s–e−ss2+4e−ss

**Answer: b**

**Explanation: V (t) = 5 + u (t) (t-5)**

**L {5 + u (t) (t-5)} = 5s + L {u (t) (t-5)}**

**= 5s+e−s L {t-4}**

**= 5s+e−s(1s2–4s)**

**= 5s+e−ss2–4e−ss.**

**14. W(t) = 2, 0≤t<4;**

**t**

^{2}, t≥4;**The Laplace transform of W (t) is ___________**

a) 2s–e−4s(2s3–8s2–14s)

b) 2s+e−4s(2s3–8s2–14s)

c) 2s–e−4s(2s3+8s2+14s)

**d) 2s+e−4s(2s3+8s2+14s)**

**Answer: d**

**Explanation: W (t) = 2 + u (t) (t**

^{2}-2)**L {2 + u (t) (t**

^{2}-2)} = 2s + L {u (t) (t^{2}-2)}**= 2s+e−4s L {(t+4)**

^{2}-2}**= 2s+e−4s L {t**

^{2}+ 8t + 14}**= 2s+e−4s(2s3+8s2+14s).**

**15. U(t) = 0, 0≤t<7;**

**(t-7)**

^{3}, t≥7;**The Laplace transform of U (t) is ___________**

**a) 6e−7ss4**

b) e−7ss4

c) 6e−7ss3

d) 3e−7ss3

**Answer: a**

**Explanation: U (t) = u (t) (t-7)**

^{3}**L {u (t) (t-7)**

^{3}} = e^{-7s}L {t^{3}}**= 3!e−7ss4=6e−7ss4.**

# Miscellaneous Examples on Fourier Series

**1. A signal g (t) = 10 sin (12πt) is ___________**

a) A periodic signal with period 6 s

b) A periodic signal with period 106 s**c) A periodic signal with period 16 s**

d) An aperiodic signal**Answer: c****Explanation: 10 sin (12πt) = A sin ωt****∴ω = 12π****∴ 2πf = 1****So, fundamental frequency = 6****Hence, fundamental period = 16 s.****2. A voltage having the waveform of a sine curve is applied across a capacitor. When the frequency of the voltage is increased, what happens to the current through the capacitor?****a) Increases**

b) Decreases

c) Remains same

d) Is zero**Answer: a****Explanation: The current through the capacitor is given by,****I _{C} = ωCV cos (ωt + 90°).**

**As the frequency is increased, I**

_{C}also increases.**3. What is the steady state value of F (t), if it is known that F(s)=2s(S+1)(s+2)(s+3)?**

a) 12

**b) 13**

c) 14

d) Cannot be determined

**Answer: b**

**Explanation: From the equation of F(s), we can infer that, a simple pole is at origin and all other poles are having negative real part.**

**∴ F(∞) = lim s→0 sF(s)**

**= lim s→0 2ss(S+1)(s+2)(s+3)**

**= 2s(S+1)(s+2)(s+3)**

**= 26=13.**

**4. What is the steady state value of F (t), if it is known that F(s) = 10(s+1)(s2+1)?**

a) -5

b) 5

c) 10

**d) Cannot be determined**

**Answer: d**

**Explanation: The steady state value of this Laplace transform is cannot be determined since; F(s) is having two poles on the imaginary axis (j and –j). Hence the answer is that it cannot be determined.**

**5. A periodic rectangular signal X (t) has the waveform as shown below. The frequency of the fifth harmonic of its spectrum is ______________**

a) 40 Hz

b) 200 Hz

c) 250 Hz

**d) 1250 Hz**

**Answer: d**

**Explanation: Periodic time = 4 ms = 4 × 10**

^{-3}**Fundamental frequency = 1034 = 250 Hz**

**∴ Frequency of the fifth harmonic = 250 × 5 = 1250 Hz.**

**6. A CRO probe has an impedance of 500 kΩ in parallel with a capacitance of 10 pF. The probe is used to measure the voltage between P and Q as shown in the figure. The measured voltage will be?**

a) 3.53 V

**b) 3.47 V**

c) 5.54 V

d) 7.00 V

**Answer: b**

**Explanation: XC=1jCω=−j2×3.14×100×103×10×10−12**

**Applying KCL at the node,**

**Va−10100+Va100+Va500+Va−j159**

**∴ V**

_{a}= 4.37∠-15.95°.**7. The Fourier series coefficient of the signal z(t) = Re{x(t)} is ____________**

a) X[k]+X[−k]2

b) X[k]−X[−k]2

**c) X[k]+X∗[−k]2**

d) X[k]−X∗[−k]2

**Answer: c**

**Explanation: Re{x (t)} = x(t)+x∗(−t)2**

**The Fourier coefficient of x* (t) is**

**X**

_{1}[k] = 1T ∫ x* (t)e^{-jkωt}dt = X∗1 [-k]**Or, X∗1 [k] = 1T ∫ x(t)e**

^{jkωt}dt = X [-k]**So, X**

_{1}[k] = X∗1 [-k]**∴ Z[k] = X[k]+X∗[−k]2.**

**8. The Fourier series expansion of a real periodic signal with fundamental frequency f**

_{0}is given by g_{p}(t) = ∑∞n=−∞Cnej2πnf0t. Given that C_{3}= 3 + j5. The value of C_{-3}is ______________a) 5 + 3j

b) -5 + 3j

c) -3 – j5

**d) 3 – j5**

**Answer: d**

**Explanation: Given that C**

_{3}= 3 + j5.**We know that for real periodic signal C**

_{-k}= C∗k**So, C**

_{-3}= C∗3 = 3 – j5.

Three- Domain Analysis of CT Systems MCQs

**9. A signal e ^{-at} sin (ωt) is the input to a real linear time invariant system. Given K and ∅ are constants, the output of the system will be of the form Ke^{-bt} sin (vt + ∅). The correct statement among the following is __________**

**a) b need not be equal to a but v must be equal to ω**

b) v need not be equal to ω but b must be equal to a

c) b must be equal to a and v must be equal to ω

d) b need not be equal to a and v need not be equal to ω

**Answer: a**

**Explanation: For a system with input e**

^{-at}sin (ωt) and output Ke^{-bt}sin (v t + ∅), frequency (v) to output must be equal to input frequency (ω) while b will depend on system parameters and need not be equal to a.**10. Let us suppose that the impulse response of a causal LTI system is given as h (t). Now, consider the following two statements:**

**Statement 1- Principle of superposition holds**

**Statement 2- h (t) = 0(for t<0)**

**Which of the following is correct?**

a) Statement 1 is correct and Statement 2 is wrong

b) Statement 2 is correct and Statement 1 is wrong

c) Both Statement 1 and Statement 2 are wrong

**d) Both statement 1 and Statement 2 are correct**

**Answer: d**

**Explanation: As we know that linear system possesses superposition theorem.**

**Hence, Statement 1 satisfies the given equation. We also know that time invariant condition depend on time.**

**Hence, Statement 2 satisfies the given equation.**

**11. The Fourier series representation of an impulse train denoted by s(t) = ∑∞n=−∞δ(t−nT0) is given by _____________**

a) 1T0∑∞n=−∞exp(−j2πntT0)

**b) 1T0∑∞n=−∞exp(−jπntT0)**

c) 1T0∑∞n=−∞exp(jπntT0)

d) 1T0∑∞n=−∞exp(j2πntT0)

**Answer: b**

**Explanation: s (t) = ∑∞n=−∞Cnejnω0t, where ω**

_{0}=(2T/T_{0})**And C**

_{n}= 1T0∫T02−T02δ(t)e−jnω0tdt**= 1.e−jnω0tT0**

**So, Fourier series representation = 1T0∑∞n=−∞exp(−jπntT0).**

**12. For which of the following a Fourier series cannot be defined?**

a) 3 sin (25t)

b) 4 cos (20t + 3) + 2 sin (710t)

**c) exp(-|t|) sin (25t)**

d) 1

**Answer: c**

**Explanation: 3 sin (25) = 25**

**4 cos (20 + 3) + 2sin (710) sum of two periodic function is also periodic function**

**For 1 which is a constant, Fourier series exists.**

**For exp (-|t|) sin (25t), due to decaying exponential decaying function, it is not periodic. So Fourier series cannot be defined for it.**

**13. The RMS value of a rectangular wave of period T, having a value of +V for a duration T**

_{1}(< T) and −V for the duration T − T_{1}= T_{2}, equals _____________**a) V**

b) V2√

c) T1−T2T V

d) T1T2 V

**Answer: a**

**Explanation: Period = T = T**

_{1}+ T_{2}**RMS value = 1T∫T0x2(t)dt−−−−−−−−−−√**

**= 1T[V2.(T1–0)+V2(T–T1)]−−−−−−−−−−−−−−−−−−−−−√**

**= V2−−−√**

**= V.**

**14. A periodic square wave is formed by rectangular pulses ranging from -1 to +1 and period = 2 units. The ratio of the power in the 7th harmonic to the power in the 5th harmonic for this waveform is equal to ____________**

a) 1

**b) 0.5**

c) 2

d) 2.5

**Answer: b**

**Explanation: For a periodic square wave nth harmonic component ∝ 1n**

**Thus the power in the nth harmonic component is ∝ 1n2**

**∴ Ratio of power in 7th harmonic to 5th harmonic for the given wage form = 172152**

**= 254 ≅ 0.5.**

**15. A useful property of the unit impulse 6 (t) is ________________**

a) 6 (at) = a 6 (t)

b) 6 (at) = 6 (t)

**c) 6 (at) = 1a 6(t)**

d) 6(at) = [6(t)]

^{a}

**Answer: c**

**Explanation: Time-scaling property of 6(t)**

**We know that by this property,**

**6(at) = 1a 6(t), a>0**

# Four**ier Series Properties – 1**

**1. How do we represent a pairing of a periodic signal with its fourier series coefficients in case of continuous time fourier series?****a) x(t) ↔ X _{n}**

b) x(t) ↔ X

_{n+1}

c) x(t) ↔ X

d) x(n) ↔ X

_{n}

**Answer: a**

**Explanation: In case of continuous time fourier series, for simplicity, we represent a pairing of a periodic signal with its fourier series coefficients as,**

**x(t) ↔ X**

_{n}**here, x(t) is the signal and Xn is the fourier series coefficient.**

**2. What are the properties of continuous time fourier series?**

a) Linearity, time shifting

b) Linearity, time shifting, frequency shifting

c) Linearity, time shifting, frequency shifting, time reversal, time scaling, periodic convolution

**d) Linearity, time shifting, frequency shifting, time reversal, time scaling, periodic convolution, multiplication, differentiation**

**Answer: d**

**Explanation: Linearity, time shifting, frequency shifting, time reversal, time scaling, periodic convolution, multiplication, differentiation are some of the properties followed by continuous time fourier series. Integration and conjugation are also followed by continuous time fourier series.**

**3. Integration and conjugation are also followed by continuous time fourier series?**

**a) True**

b) False

**Answer: a**

**Explanation: Linearity, time shifting, frequency shifting, time reversal, time scaling, periodic convolution, multiplication, differentiation are some of the properties followed by continuous time fourier series. Integration and conjugation are also followed by continuous time fourier series.**

**4. If x(t) and y(t) are two periodic signals with coefficients X**

_{n}and Y_{n}then the linearity is represented as?**a) ax(t) + by(t) = aX**

_{n}+ bY_{n}b) ax (t) + by(t) = X

_{n}+ bY

_{n}

c) ax(t) + by(t) = aX

_{n}+ Y

_{n}

d) ax(t) + by(t) = X

_{n}+ Y

_{n}

**Answer: a**

**Explanation: ax(t) + by(t) = aX**

_{n}+ bY_{n}, x(t) and y(t) are two periodic signals with coefficients X_{n}and Y_{n}.**5. How is time shifting represented in case of periodic signal?**

a) If x(t) is shifted to t

_{0}, X

_{n}is shifted to t

_{0}

b) x(t-t

_{0}), Y

_{n}= X

_{n}e

^{-njwt0}

**c) X**

_{n}= x(t-t_{0}), Y_{n}= X_{n}e^{-njwt0}d) X

_{n}= x(-t

_{0}), Y

_{n}= X

_{n}e

^{-njwt0}

**Answer: c**

**Explanation: If x(t) and y(t) are two periodic signals with coefficients X**

_{n}and Y_{n}, then if a signal is shifted to t_{0}, then the property says,**X**

_{n}= x(t-t_{0}), Y_{n}= X_{n}e^{-njwt0}**6. What is the frequency shifting property of continuous time fourier series?**

a) Multiplication in the time domain by a real sinusoid

**b) Multiplication in the time domain by a complex sinusoid**

c) Multiplication in the time domain by a sinusoid

d) Addition in the time domain by a complex sinusoid

**Answer: b**

**Explanation: If x(t) and y(t) are two periodic signals with coefficients Xn and Yn,**

**Then y(t)= e**

^{jmwt}x(t)↔Y_{n}=X_{n-m}.**Hence, we can see that a frequency shift corresponds to multiplication in the time domain by complex sinusoid whose frequency is equal to the time shift.**

**7. What is the time reversal property of fourier series coefficients?**

**a) Time reversal of the corresponding sequence of fourier series**

b) Time reversal of the last term of fourier series

c) Time reversal of the corresponding term of fourier series

d) Time reversal of the corresponding sequence

**Answer: a**

**Explanation: x(t)↔ X**

_{n}**Y(t) = x(-t)↔Y**

_{n}=X_{-n}.**That is the time reversal property of fourier series coefficients is time reversal of the corresponding sequence of fourier series.**

**8. It does not depend whether the signal is odd or even, it is always reversal of the corresponding sequence of fourier series.**

a) True

**b) False**

**Answer: b**

**Explanation: It does depend whether the signal is odd or even.**

**If the signal is even, the reversal is positive and if the signal is odd, the reversal is negative.**

**9. Why does the signal change while time scaling?**

**a) Because the frequency changes**

b) Time changes

c) Length changes

d) Both frequency and time changes

**Answer: a**

**Explanation: x(t)↔X**

_{n}**Y(t) = x(at)↔Y**

_{n}= X_{n}**Hence, the fourier coefficients have not changed but the representation has changed because of changes in fundamental frequency.**

**10. What is the period of the signal when it is time shifted?**

a) Changes according to the situation

b) Different in different situation

**c) Remains the same**

d) Takes the shifted value

**Answer: c**

**Explanation: The period of the periodic signal does not change even if it is time shifted.**

**If x(t) and y(t) are two periodic signals with coefficients X**

_{n}and Y_{n}, then if a signal is shifted to t_{0}, then the property says,**X**

_{n}= x(t-t_{0}), Y_{n}= X_{n}e^{-njwt0}.# Fourier Series Properties – 2

**1. Can continuous time fourier series undergo periodic convolution?**

a) They cannot undergo periodic convoluion

b) They can undergo in certain situations**c) They undergo periodic convolution**

d) Only even signals undergo periodic convolution**Answer: c****Explanation: Continuous time fourier series undergoes periodic convolution.****X(t)*y(t)=z(t) ↔ X _{n}Y_{n} = Z_{n}.**

**2. What is the outcome of a periodic convolution of signals in case of continuous time fourier series?**

a) Division in frequency domain

**b) Multiplication in frequency domain**

c) Convolution is easier

d) Addition of signals in frequency domain

**Answer: b**

**Explanation: This is a very important property of continuous time fourier series, it leads to the conclusion that the outcome of a periodic convolution is the multiplication of the signals in frequency domain representation.**

**X(t)*y(t)=z(t) ↔ X**

_{n}Y_{n}=Z_{n}.**3. What is the multiplication property of continuous time fourier series?**

**a) Convolution of the signals**

b) Multiplication of the elements of the signal

c) Division of the frequency domain

d) Addition of the signals in frequency domain

**Answer: a**

**Explanation: In the case of continuous time fourier series, the multiplication property leads to discrete time convolution of the signals.**

**z(t)=x(t)y(t) ↔ Z**

_{n}= X_{n}Y_{n-k}.**4. What is the differentiation property of continuous time fourier series?**

a) Y

_{n}= jnwtX

_{n}

b) Y

_{n}= jntX

_{n}

**c) Y**

_{n}= jnwX_{n}d) X

_{n}= jnwtX

_{n}

**Answer: c**

**Explanation: x(t) ↔X**

_{n}, x(t) is the signal and X_{n}is the coefficient.**Then, Y**

_{n}= jnwX_{n}.**5. What is the fourier series coefficient for n=0?**

**a) Zero**

b) Unity

c) Depends on the situation

d) Non zero positive

**Answer: a**

**Explanation: The differentiation property of the continuous time fourier series is,**

**Y(t) = dx(t)/dt ↔ Y**

_{n}= jnwX_{n}.**Hence, the differentiation property of time averaged value of the differentiated signal to be zero, hence, fourier series coefficient for n=0 is zero.**

**6. What is the integration property of the continuous time fourier series?**

**a) y(t) ↔ Y**

_{n}= 1/jnwX_{n}b) y(t) ↔ Y

_{n}= 1/jwX

_{n}

c) y(t) ↔ Y

_{n}= 1/jnX

_{n}

d) y(t) ↔ Y

_{n}= 1/jnw

**Answer: a**

**Explanation: y(t)↔ Y**

_{n}= 1/jnwX_{n}, here x(t) is the signal and y(t) is the output.**This is the integration property of the signal.**

**7. What is the smoothing operation?**

a) Differentiation property

b) Multiplication property

**c) Integration property**

d) Conjugation property

**Answer: c**

**Explanation: The integration attenuates the magnitude of the high frequency components of the signal. High frequency contributors cause sharp details such as occurring at the points of discontinuity. Hence, integration smoothens the signal, hence it is called a smoothening operation.**

**8. What is the complex conjugate property of a fourier series?**

a) It leads to convolution

**b) It leads to time reversal**

c) It leads to multiplication

d) It leads to addition of signals

**Answer: b**

**Explanation: x(t) ↔ X**

_{n}**Y(t) = *x(t) ↔Y**

_{n}=*X_{-n}**It leads to time reversal.**

**9. If the signal x(t) is odd, what will be the fourier series soeffiients?**

**a) Real and even**

b) Odd

c) Real only

d) Real and odd

**Answer: a**

**Explanation: If the signal is real and odd, the fourier series coefficients are conjugate symmetric.**

**And its fourier series coefficients are real and even.**

**X**

_{n}= X_{-n}*= X_{n}.**10. If the signal x(t) is even, what will be the fourier series coefficients?**

a) Real and even

b) Odd

c) Real only

**d) Imaginary and odd**

**Answer: d**

**Explanation: If the signal is real and even, the fourier series coefficients are conjugate symmetric.**

**And its fourier series coefficients are imaginary and even.**

**X**

_{n}= X_{-n}*= -X_{n}.**Most Up To Date Signals and Systems ****MCQs – ****Fourier Series MCQs ( Signals and Systems ) MCQs**