MATH 416 — Applied Mathematics (BSU)
Week 1 — Periodic Functions & Fourier Coefficients (Complex Form)
Purpose, Context & Signal Motivation (complex-only)
We model periodic signals using the orthonormal family of complex exponentials \(e_n(x)=e^{2\pi i n x}\) on \([0,1]\).
A 1-periodic signal \(f\) is represented formally by the Fourier expansion
\(f(x)\sim\sum_{n\in\mathbb Z}\widehat f(n)\,e^{2\pi i n x}\),
where the coefficients \(\widehat f(n)\) are obtained by projecting \(f\) onto \(e_n\).
Learning outcomes. Define 1-periodicity; compute \(\widehat f(n)=\int_0^1 f(x)e^{-2\pi i n x}\,dx\);
use orthonormality \(\langle e_m,e_n\rangle=\delta_{mn}\); apply conjugate symmetry for real \(f\); and exploit translation/modulation rules.
use orthonormality \(\langle e_m,e_n\rangle=\delta_{mn}\); apply conjugate symmetry for real \(f\); and exploit translation/modulation rules.
Conventions & Normalization (unitary, complex)
- Period & fundamental interval. \(f\) is \(T\)-periodic if \(f(x+T)=f(x)\). We work on any interval of length \(T\).
- Course default. Normalize to \(T=1\) with inner product \(\langle f,g\rangle=\int_0^1 f(x)\overline{g(x)}\,dx\).
Then \(\{e_n(x)=e^{2\pi i n x}\}_{n\in\mathbb Z}\) is orthonormal. - Coefficients & reconstruction. \(\widehat f(n)=\langle f,e_n\rangle\), and formally \(f\sim\sum_{n\in\mathbb Z}\widehat f(n)e^{2\pi i n x}\).
- Parseval/Plancherel. \(\displaystyle \int_0^1 |f(x)|^2\,dx=\sum_{n\in\mathbb Z}|\widehat f(n)|^2\).
Period \(T\) version (recommended unitary form).
Use \(\displaystyle \langle f,g\rangle_T=\frac{1}{T}\int_0^T f(t)\overline{g(t)}\,dt\), basis \(e_n^T(t)=e^{2\pi i n t/T}\),
coefficients \(\displaystyle C_n=\frac{1}{T}\int_0^T f(t)e^{-2\pi i n t/T}\,dt\), and \(f(t)\sim\sum_{n\in\mathbb Z}C_n e^{2\pi i n t/T}\).
Use \(\displaystyle \langle f,g\rangle_T=\frac{1}{T}\int_0^T f(t)\overline{g(t)}\,dt\), basis \(e_n^T(t)=e^{2\pi i n t/T}\),
coefficients \(\displaystyle C_n=\frac{1}{T}\int_0^T f(t)e^{-2\pi i n t/T}\,dt\), and \(f(t)\sim\sum_{n\in\mathbb Z}C_n e^{2\pi i n t/T}\).
Quick Reference (copy into your notebook)
- 1-periodic: \(f(x+1)=f(x)\).
- Mean over one period: \(\displaystyle \int_0^1 f(x)\,dx=\widehat f(0)\).
- Orthonormal atoms: \(\langle e_m,e_n\rangle=\int_0^1 e^{2\pi i m x}\overline{e^{2\pi i n x}}\,dx=\delta_{mn}\).
- Fourier coefficient: \(\displaystyle \boxed{\ \widehat f(n)=\int_0^1 f(x)\,e^{-2\pi i n x}\,dx\ }\).
- Series (formal): \(f(x)\sim\sum_{n\in\mathbb Z}\widehat f(n)e^{2\pi i n x}\).
- Real signals: If \(f\) is real-valued, then \(\widehat f(-n)=\overline{\widehat f(n)}\).
All formulas are for \([0,1]\) and 1-periodic extensions. For period \(T\), replace \(2\pi n x\) by \(2\pi n x/T\) and use the unitary factors above.
Signals → Periodicity → Inner Products → Orthonormal Exponentials
1. Signals & periodicity
Daylight intensity (toy model). Let \(I(t)\) denote daily light intensity. A 1-periodic model is
\(I(t)\approx \widehat I(0)+\widehat I(1)e^{2\pi i t}+\widehat I(-1)e^{-2\pi i t}\) with \(\widehat I(-1)=\overline{\widehat I(1)}\) if \(I\) is real-valued.
\(I(t)\approx \widehat I(0)+\widehat I(1)e^{2\pi i t}+\widehat I(-1)e^{-2\pi i t}\) with \(\widehat I(-1)=\overline{\widehat I(1)}\) if \(I\) is real-valued.
2. Geometry via the inner product
Similarity is measured by \(\langle f,g\rangle=\int_0^1 f(x)\overline{g(x)}\,dx\). Orthogonality means \(\langle f,g\rangle=0\).
The exponentials \(\{e_n\}\) are orthonormal, so coefficients are obtained by projection: \(\widehat f(n)=\langle f,e_n\rangle\).
3. Formal reconstruction
If \(f\in L^2([0,1])\), then \(f\) has Fourier coefficients \(\widehat f(n)\) with \(\sum|\widehat f(n)|^2<\infty\) and
(in \(L^2\) sense) \(f=\sum_{n\in\mathbb Z}\widehat f(n)e^{2\pi i n x}\).
Worked Examples — Computing \(\widehat f(n)\)
Example 1 — DC + single frequency
\(f(x)=I_0 + A\,e^{2\pi i x} + \overline{A}\,e^{-2\pi i x}\) (real-valued if \(A\in\mathbb C\) with the conjugate term).
Then \(\widehat f(0)=I_0\), \(\widehat f(1)=A\), \(\widehat f(-1)=\overline{A}\), and \(\widehat f(n)=0\) for \(|n|\ge2\).
Then \(\widehat f(0)=I_0\), \(\widehat f(1)=A\), \(\widehat f(-1)=\overline{A}\), and \(\widehat f(n)=0\) for \(|n|\ge2\).
Example 2 — Phase shift
\(f(x)=\frac12 e^{2\pi i (x-a)}+\frac12 e^{-2\pi i (x-a)}\) (a cosine with phase \(a\)).
Then \(\widehat f(1)=\tfrac12 e^{-2\pi i a}\), \(\widehat f(-1)=\tfrac12 e^{2\pi i a}\), others \(0\).
Then \(\widehat f(1)=\tfrac12 e^{-2\pi i a}\), \(\widehat f(-1)=\tfrac12 e^{2\pi i a}\), others \(0\).
Example 3 — Square wave
\(f(x)=\chi_{[0,1/2)}(x)-\chi_{[1/2,1)}(x)\) (1-periodic). Then
\[
\widehat f(n)=\frac{2}{\pi i n}\big(1-(-1)^n\big)=
\begin{cases}\displaystyle \frac{4}{\pi i\,n},& n\ \text{odd},\\[6pt] 0,& n\ \text{even},\end{cases}
\quad \widehat f(0)=0.
\]
\[
\widehat f(n)=\frac{2}{\pi i n}\big(1-(-1)^n\big)=
\begin{cases}\displaystyle \frac{4}{\pi i\,n},& n\ \text{odd},\\[6pt] 0,& n\ \text{even},\end{cases}
\quad \widehat f(0)=0.
\]
Example 4 — Sawtooth wave
\(f(x)=x-\tfrac12\) on \([0,1]\) (1-periodic). Then \(\widehat f(0)=0\) and, for \(n\neq 0\),
\[
\widehat f(n)=-\frac{1}{2\pi i\,n}=\frac{i}{2\pi\,n}.
\]
\[
\widehat f(n)=-\frac{1}{2\pi i\,n}=\frac{i}{2\pi\,n}.
\]
Example 5 — Triangle wave (even about \(x=\tfrac12\))
\(f(x)=1-4\,|x-\tfrac12|\) on \([0,1]\) (1-periodic). Then \(\widehat f(0)=0\), \(\widehat f(2k)=0\), and for odd \(n=2k+1\),
\[
\widehat f(\pm n)=\frac{4}{\pi^2 n^2}\ \ (\text{real}).
\]
\[
\widehat f(\pm n)=\frac{4}{\pi^2 n^2}\ \ (\text{real}).
\]
Example 6 — From samples to coefficients (DFT viewpoint)
For samples \(x_m=\frac{m}{N}\), \(m=0,\dots,N-1\), a Riemann-sum approximation yields
\[
\widehat f(n)\approx \frac{1}{N}\sum_{m=0}^{N-1} f(x_m)\,e^{-2\pi i n x_m},\qquad n=0,\dots,N-1.
\]
\[
\widehat f(n)\approx \frac{1}{N}\sum_{m=0}^{N-1} f(x_m)\,e^{-2\pi i n x_m},\qquad n=0,\dots,N-1.
\]
Meeting 1 — Periodicity, inner product, orthogonality (complex form)
1. Periodicity & one-period integrals
\(f:\mathbb R\to\mathbb C\) is 1-periodic if \(f(x+1)=f(x)\) for all \(x\).
One-period integrals are translation-invariant:
\[
\int_0^1 f(x-y)\,dx=\int_0^1 f(x)\,dx\quad (y\in\mathbb R).
\]
One-period integrals are translation-invariant:
\[
\int_0^1 f(x-y)\,dx=\int_0^1 f(x)\,dx\quad (y\in\mathbb R).
\]
Make \(g:[0,1]\to\mathbb C\) 1-periodic via \(f(x)=g(x-\lfloor x\rfloor)\).
2. Orthonormality of exponentials
\(\displaystyle \langle e_m,e_n\rangle = \int_0^1 e^{2\pi i (m-n) x}\,dx=\delta_{mn}\).
3. Coefficients and projection
\[
\boxed{\ \widehat f(n)=\int_0^1 f(x)\,e^{-2\pi i n x}\,dx\ }\quad (n\in\mathbb Z),\qquad
f(x)\sim\sum_{n\in\mathbb Z}\widehat f(n)\,e^{2\pi i n x}.
\]
\boxed{\ \widehat f(n)=\int_0^1 f(x)\,e^{-2\pi i n x}\,dx\ }\quad (n\in\mathbb Z),\qquad
f(x)\sim\sum_{n\in\mathbb Z}\widehat f(n)\,e^{2\pi i n x}.
\]
Activities
A1. Compute \(\int_0^1 e^{2\pi i n x}\,dx\) for \(n\in\mathbb Z\).
A2. Show \(\int_0^1 e^{2\pi i m x}\overline{e^{2\pi i n x}}\,dx=\delta_{mn}\).
A3. For \(f=\chi_{[0,1/2)}-\chi_{[1/2,1)}\), derive the closed form for \(\widehat f(n)\).
Exit slip. State the definition of \(\widehat f(n)\) and whether it depends on \(x\).
Meeting 2 — Properties of \(\widehat f\): symmetry, translation, modulation
1. Conjugate symmetry (real signals)
If \(f\) is real-valued, then \(\widehat f(-n)=\overline{\widehat f(n)}\).
2. Translation & modulation
If \(g(x)=f(x-a)\) (time shift), then \(\widehat g(n)=e^{-2\pi i n a}\widehat f(n)\).
If \(h(x)=e^{2\pi i k x} f(x)\) (modulation), then \(\widehat h(n)=\widehat f(n-k)\).
If \(h(x)=e^{2\pi i k x} f(x)\) (modulation), then \(\widehat h(n)=\widehat f(n-k)\).
3. Two computations for \(\widehat f(n)\)
B1. \(f(x)=x(1-x)\) on \([0,1]\) (1-periodic). Compute \(\widehat f(n)\) by integrating by parts.
B2. Square wave of Example 3. Verify \(\widehat f(n)=\dfrac{2}{\pi i n}\big(1-(-1)^n\big)\).
4. Smoothness vs. decay (heuristic)
The smoother \(f\) is, the faster \(|\widehat f(n)|\) tends to \(0\) as \(|n|\to\infty\).
Jump discontinuities produce \(O(1/|n|)\) tails and Gibbs oscillations near the jumps.
Jump discontinuities produce \(O(1/|n|)\) tails and Gibbs oscillations near the jumps.
Exit slip. State the translation and modulation rules for \(\widehat f\).
Style & Template Credit
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