Fourier series

The Fourier series represents a periodic signal of period $T$ as a sum of complex sinusoids at integer multiples of the fundamental frequency $f_0=1/T$:

Synthesis (signal from coefficients):

$x(t)={\sum }_{k=-\infty }^{\infty }{c}_x[k]e^{j2\pi kt/T}.$

Analysis (coefficients from signal):

$c_x[k]=\frac{1}{T}{\int }_{T}x(t)e^{-j2\pi kt/T}dt,$

where ${\int }_T$ means “integrate over any one period.”

These two equations are the heart of the chapter. They are not on the formula sheet; you have to know them by heart. Everything else (properties, pairs, conversions) is on the sheet.

Reading the formulas

The $k$-th coefficient $c_x[k]$ describes the strength of the harmonic at frequency $k{f}_0=k/T$. So $k=1$ is the fundamental, $k=2$ is the second harmonic, etc. Negative $k$ values correspond to complex sinusoids spinning the opposite direction — they pair with positive $k$ to make real-valued sinusoids when $x$ is real (see Conjugate symmetry of Fourier coefficients).

The $k=0$ coefficient is special: it’s the DC component (average value over one period), since $e^{j\cdot 0}=1$. So $c_x[0]=(1/T)\int x(t)dt$.

Why it works

The derivation depends on the orthogonality of complex sinusoids at integer-multiple frequencies. Multiply the synthesis equation by $e^{-j2\pi qt/T}$ and integrate over one period: by orthogonality, only the $k=q$ term survives, and you get the analysis formula.

Trigonometric form

For real signals, an equivalent form uses real cosines and sines:

$x(t)=a_x[0]+{\sum }_{k=1}^{\infty }\left\{a_x[k]\cos (2\pi kt/T)+b_x[k]\sin (2\pi kt/T)\right\}.$

The complex and trigonometric coefficients are related by

$c_x[k]=\frac{a_x[k]-j{b}_x[k]}{2},c_x[-k]=c_x^{\ast }[k]=\frac{a_x[k]+j{b}_x[k]}{2}(k\ge 1),$

with $c_x[0]=a_x[0]$. The factor of 2 is the most common source of error in converting between forms. See Trigonometric form of Fourier series for the conversion details.

Standard pairs

The formula sheet lists Fourier-series pairs for common building blocks. The most useful:

• $e^{j2\pi t/T_0}\leftrightarrow \delta [k-m]$, representation period $m{T}_0$.
• $\cos (2\pi qt/T_0)\leftrightarrow \frac{1}{2}(\delta [k+qm]+\delta [k-qm])$, period $m{T}_0$.
• $\sin (2\pi qt/T_0)\leftrightarrow \frac{j}{2}(\delta [k+qm]-\delta [k-qm])$, period $m{T}_0$.
• $1\leftrightarrow \delta [k]$ — a constant has only a DC coefficient.
• ${\delta }_{T_0}(t)\leftrightarrow (1/T_0){\delta }_m[k]$ — an impulse train has equal coefficients at every harmonic that’s a multiple of $m$.
• $\mathrm{rect}(t/w)\ast {\delta }_{T_0}(t)\leftrightarrow (w/T_0)\mathrm{sinc}(wk/(m{T}_0)){\delta }_m[k]$ — a periodic train of rectangles becomes a sinc-sampled spectrum.
• $\mathrm{tri}(t/w)\ast {\delta }_{T_0}(t)\leftrightarrow (w/T_0){\mathrm{sinc}}^2(wk/(m{T}_0)){\delta }_m[k]$ — a periodic train of triangles.

The pattern: any periodic train made by convolving a pulse with ${\delta }_{T_0}$ has coefficients $(w/T_0)$ times a sinc-like function of $k$, masked by the Kronecker delta ${\delta }_m[k]$ (zeroing indices not multiples of $m$).

Standard properties

All on the formula sheet, but worth knowing what each says:

• Linearity: $\alpha x+\beta y\leftrightarrow \alpha c_x+\beta c_y$.
• Time-shift: $x(t-t_0)\leftrightarrow e^{-j2\pi k{t}_0/T}{c}_x[k]$. The magnitude is unchanged; the phase rotates linearly in $k$.
• Frequency-shift: $e^{j2\pi k_{0}t/T}x(t)\leftrightarrow c_x[k-k_0]$. The dual of time-shift; foundation of amplitude modulation.
• Time differentiation: $dx/dt\leftrightarrow (j2\pi k/T)c_x[k]$. Differentiation becomes algebraic multiplication.
• Multiplication–convolution duality: $x(t)y(t)\leftrightarrow c_x[k]\ast c_y[k]$ (discrete convolution). Periodic-convolution-in-time corresponds to multiplication-in-$k$ (with a factor of $T$).
• Change of period: representing $x$ over $mT$ instead of $T$ adds zero coefficients between the existing harmonics.

Parseval’s theorem

$\frac{1}{T}{\int }_T|x(t){|}^{2}dt={\sum }_{k=-\infty }^{\infty }|c_x[k]{|}^2.$

The average power in the time domain equals the sum of squared magnitudes in the frequency domain. Useful for sanity-checking calculations and for estimating signal power from a few dominant coefficients. See Parseval’s theorem.

Convergence

For any periodic signal that is absolutely integrable over a period, has finitely many maxima/minima per period, and has finitely many discontinuities of bounded jump size per period — the Dirichlet conditions — the synthesis equation converges to $x(t)$ at every point of continuity and to the midpoint of the left and right limits at jump points.

Two different limits to keep apart. The full infinite series converges pointwise at every point: to $x(t)$ where $x$ is continuous, to the midpoint of the left/right limits at jumps. The truncated partial sum (the sum of just the first $N$ harmonics) behaves differently near jumps: it overshoots and undershoots the signal by about 9% of the jump size, and the overshoot doesn’t shrink as $N\to \infty$ — it just gets narrower (the Gibbs phenomenon). So the series and its finite truncations agree everywhere away from jumps but diverge in behavior near them: pointwise convergence holds, uniform convergence fails.

Where this leads

The Fourier series handles only periodic signals. For aperiodic signals, take $T\to \infty$ and let the discrete sum become a continuous integral — this is the Fourier transform, and the leap is one of the central computations in the next chapter.

Why this matters for circuits (Electronics I)

This is the result that justifies frequency-domain thinking in electronics. Any repeating signal — no matter how jagged — is a sum of sinusoids; an aperiodic one is an integral over a continuum of them via the Fourier transform. Since a biased amplifier or filter is linear in the small-signal regime, its effect on a complicated signal is completely determined by what it does to a single sinusoid at each frequency. That is precisely why a circuit’s frequency response, its bandwidth, and the action of a filter are meaningful, sufficient descriptions of how it processes every possible input.

Any signal decomposes into a sum of sinusoids (Fourier series); aperiodic signals use the Fourier transform.