Properties of Fourier Series

Introduction

Just as the Laplace transform has properties that simplify circuit analysis, the Fourier series has a set of powerful properties that allow us to manipulate and analyze periodic signals without recomputing integrals from scratch every time. These properties are your tools for efficiency — knowing them means you can derive the Fourier series of complex signals from simpler known results through operations like shifting, scaling, differentiation, and multiplication.

For B.Tech students, mastering these properties is essential for exam problem-solving and for understanding how operations in the time domain affect the frequency content of a signal. Each property tells you something physically meaningful about the relationship between time-domain modifications and their spectral consequences.

Notation Convention

Let $x(t)$ be a periodic signal with period $T_0$, fundamental frequency $\omega_0 = 2\pi/T_0$, and exponential Fourier coefficients $c_n$. We write: $x(t) \xleftrightarrow{FS} c_n$

Linearity

If $x_1(t) \xleftrightarrow{FS} a_n$ and $x_2(t) \xleftrightarrow{FS} b_n$ (same period), then:

$$\alpha x_1(t) + \beta x_2(t) \xleftrightarrow{FS} \alpha a_n + \beta b_n$$

This means the Fourier coefficients of a linear combination are the same linear combination of the individual coefficients. It works because integration is linear.

Example: If we know the Fourier series of a square wave and a triangular wave (same period), we can immediately write the series for any weighted sum of them.

Time Shifting

If $x(t) \xleftrightarrow{FS} c_n$, then:

$$x(t - t_0) \xleftrightarrow{FS} c_n e^{-jn\omega_0 t_0}$$

Shifting a signal in time multiplies its Fourier coefficients by a complex exponential (a linear phase factor). The magnitude spectrum $|c_n|$ is unchanged — only the phase shifts.

Physical meaning: Delaying a signal doesn't change what frequencies are present or how strong they are — it only changes the relative phase of each component. The $n$th harmonic acquires a phase shift of $-n\omega_0 t_0$ (proportional to harmonic number).

Example: A pulse train shifted by $T_0/4$ has the same magnitude spectrum, but each coefficient gets multiplied by $e^{-jn\pi/2}$.

Frequency Shifting (Modulation)

$$x(t)e^{jm\omega_0 t} \xleftrightarrow{FS} c_{n-m}$$

Multiplying by a complex exponential at the $m$th harmonic shifts all coefficients by $m$ positions. This is the basis of frequency-domain shifting (modulation).

For real modulation with a cosine carrier:

$$x(t)\cos(m\omega_0 t) \xleftrightarrow{FS} \frac{1}{2}(c_{n-m} + c_{n+m})$$

Each coefficient splits into two copies shifted to $n+m$ and $n-m$ — this is exactly what amplitude modulation does in communications!

Time Reversal

$$x(-t) \xleftrightarrow{FS} c_{-n}$$

Time reversal reflects the coefficients around $n = 0$. For real signals where $c_{-n} = c_n^*$, this means: $x(-t) \xleftrightarrow{FS} c_n^*$ — time reversal conjugates the coefficients.

Conjugation

$$x^*(t) \xleftrightarrow{FS} c_{-n}^*$$

For real signals, $x^*(t) = x(t)$, confirming $c_{-n}^* = c_n$ (conjugate symmetry).

Time Scaling

If $x(t)$ has period $T_0$ with coefficients $c_n$, then $x(at)$ has period $T_0/|a|$ with the same coefficients $c_n$ but at a higher fundamental frequency $a\omega_0$.

The spectral content is the same, but the harmonic spacing is compressed or expanded.

Differentiation

$$\frac{dx(t)}{dt} \xleftrightarrow{FS} jn\omega_0 c_n$$

Differentiation multiplies the $n$th coefficient by $jn\omega_0$. This means:

• DC component ($n=0$) is eliminated (derivative of a constant is zero)
• Higher harmonics are amplified by factor $n$ (differentiation emphasizes high frequencies)
• Phase shifts by $+90°$ (multiplication by $j$)

This property is why differentiators are high-pass filters and why differentiation amplifies noise.

Generalization: $\frac{d^k x}{dt^k} \xleftrightarrow{FS} (jn\omega_0)^k c_n$

Integration

$$\int_{-\infty}^{t} x(\tau)d\tau \xleftrightarrow{FS} \frac{c_n}{jn\omega_0} \quad (n \neq 0, \text{ and } c_0 = 0)$$

Integration divides the $n$th coefficient by $jn\omega_0$:

• Higher harmonics are attenuated by factor $1/n$ (integration smooths signals)
• Phase shifts by $-90°$
• This requires $c_0 = 0$ (the average of $x(t)$ must be zero, otherwise the integral grows unboundedly)

Integration is a low-pass operation — it suppresses high frequencies.

Multiplication (Windowing)

If $x(t) \xleftrightarrow{FS} a_n$ and $y(t) \xleftrightarrow{FS} b_n$ (same period), then:

$$x(t) \cdot y(t) \xleftrightarrow{FS} \sum_{k=-\infty}^{\infty} a_k b_{n-k}$$

Multiplication in time corresponds to convolution of coefficients (discrete convolution of the two spectra). This is the dual of the convolution theorem.

Convolution

If $x(t)$ and $y(t)$ have the same period $T_0$, their periodic convolution:

$$z(t) = \frac{1}{T_0}\int_0^{T_0} x(\tau)y(t-\tau)d\tau \xleftrightarrow{FS} a_n \cdot b_n$$

Periodic convolution in time corresponds to multiplication of coefficients. This is why filtering a periodic signal simply scales each harmonic by the filter's frequency response at that harmonic.

Parseval's Theorem (Power Conservation)

$$P = \frac{1}{T_0}\int_0^{T_0} |x(t)|^2 dt = \sum_{n=-\infty}^{\infty} |c_n|^2$$

The total average power equals the sum of powers in all harmonics. This is energy conservation in the frequency domain.

For the trigonometric form:

$$P = \left(\frac{a_0}{2}\right)^2 + \frac{1}{2}\sum_{n=1}^{\infty}(a_n^2 + b_n^2) = \frac{a_0^2}{4} + \frac{1}{2}\sum_{n=1}^{\infty}A_n^2$$

Worked Examples

Example 1: Given a square wave with $c_n = \frac{2}{jn\pi}$ (odd $n$ only), find the coefficients of its derivative.

Solution: By differentiation property: $c_n' = jn\omega_0 \cdot \frac{2}{jn\pi} = \frac{2\omega_0}{\pi}$ for odd $n$.

The derivative of a square wave is a train of impulses (positive at rising edges, negative at falling edges).

Example 2: If $x(t) \xleftrightarrow{FS} c_n$, find the coefficients of $x(t-T_0/4)$.

Solution: $c_n' = c_n e^{-jn\omega_0 T_0/4} = c_n e^{-jn\pi/2}$

For $n=1$: multiply by $e^{-j\pi/2} = -j$. For $n=2$: multiply by $e^{-j\pi} = -1$.

Example 3: Verify Parseval's theorem for $x(t) = \cos(\omega_0 t)$.

Time domain: $P = \frac{1}{T_0}\int_0^{T_0}\cos^2(\omega_0 t)dt = 1/2$

Frequency domain: $c_1 = 1/2, c_{-1} = 1/2$. Sum: $|1/2|^2 + |1/2|^2 = 1/4 + 1/4 = 1/2$ ✓

Summary Table

Key Takeaways

• Time shift → phase rotation of coefficients (magnitude unchanged)
• Differentiation → multiply by $jn\omega_0$ (high-pass behavior)
• Integration → divide by $jn\omega_0$ (low-pass behavior)
• Time multiplication → coefficient convolution; time convolution → coefficient multiplication
• Parseval's theorem ensures power conservation between time and frequency representations
• These properties allow derivation of new Fourier series from known ones without recomputing integrals