Chapter 17, Problem 1RQ

Which of the following cannot be a Fourier series?

1. (a) $t-\frac{t^2}{2}+\frac{t^3}{3}-\frac{t^4}{4}+\frac{t^5}{5}$
2. (b) 5 sin t + 3 sin 2t − 2 sin 3t + sin 4t
3. (c) sin t − 2 cos 3t + 4 sin 4t + cos 4t
4. (d) sin t + 3 sin 2.7t − cos πt + 2 tan πt
5. (e) $1+e^{-j\pi t}+\frac{e^{-j2\pi t}}{2}+\frac{e^{-j3\pi t}}{3}$

To determine

Choose the correct option that cannot be a Fourier series.

Answer to Problem 1RQ

The correct option is $\underset{\_}{\left(\text{a}\right)t-\frac{t^2}{2}+\frac{t^3}{3}-\frac{t^4}{4}+\frac{t^5}{5}}$.

Explanation of Solution

Discussion:

The signal is said to be a periodic function when it repeats for every $T$ seconds. The periodic function $f\left(t\right)$ satisfies the following condition,

$f\left(t\right)=f\left(t+nT\right)$

Here,

$n$ is an integer, and

$T$ is the period of the function.

The continuous time signals such as cosine and sinusoidal function satisfies the condition of periodic functions.

According to the Fourier theorem, the periodic function of the angular frequency $\left({\omega }_0\right)$ can be defined as an infinite sum of sinusoidal or cosine functions that are integral multiples of angular frequency ${\omega }_0$.

In other words, the Fourier series function $f\left(t\right)$ is a representation that resolves the function $f\left(t\right)$ into a dc component and ac component that comprises with an infinite series of harmonic sinusoids.

The trigonometric Fourier series of function $f\left(t\right)$ is expressed as,

$f\left(t\right)=a_0+{\displaystyle \sum _{n=1}^{\infty }\left(a_n\cos n{\omega }_{0}t+b_n\sin n{\omega }_{0}t\right)}$

Here,

$a_0$ is the dc component of $f\left(t\right)$,

$a_n$ and $b_n$ are the Fourier coefficients, and

${\omega }_0$ is the angular frequency.

Option (a):

Given that,

$f\left(t\right)=t-\frac{t^2}{2}+\frac{t^3}{3}-\frac{t^4}{4}+\frac{t^5}{5}$        (1)

Refer to equation (1), the function $f\left(t\right)$ doesn’t include the continuous time signals to satisfy the periodic function conditions.

Therefore, it is not a Fourier series.

Option (b):

Given that,

$f\left(t\right)=5\sin t+3\sin 2t-2\sin 3t+\sin 4t$        (2)

Refer to equation (2), the function $f\left(t\right)$ comprises of sum of sinusoidal functions with angular frequency ${\omega }_0$.

Therefore, it is a Fourier series.

Option (c):

Given that,

$f\left(t\right)=\sin t-2\cos 3t+4\sin 4t+\cos 4t$        (3)

Refer to equation (3), the function $f\left(t\right)$ comprises of sum of sinusoidal and cosine functions with angular frequency ${\omega }_0$.

Therefore, it is a Fourier series.

Option (d):

Given that,

$f\left(t\right)=\sin t+3\sin 2.7t-\cos \pi t+2\tan \pi t$        (4)

Refer to equation (4), the function $f\left(t\right)$ comprises of sum of sinusoidal and cosine functions with angular frequency ${\omega }_0$.

Therefore, it is a Fourier series.

Option (e):

Given that,

$f\left(t\right)=1+e^{-j\pi t}+\frac{e^{-j2\pi t}}{2}+\frac{e^{-j3\pi t}}{3}$        (5)

Refer to equation (5), the function $f\left(t\right)$ comprises of dc component and exponential function that includes the sum of sinusoidal and cosine functions with angular frequency ${\omega }_0$.

Therefore, it is a Fourier series.

From above, it is clear that the option (a) is correct and the options (b), (c), (d) and (e) are incorrect.

Conclusion:

Thus, the correct option is $\underset{\_}{\left(\text{a}\right)t-\frac{t^2}{2}+\frac{t^3}{3}-\frac{t^4}{4}+\frac{t^5}{5}}$.