Kyoto-University
京都大学 情報学研究科 知能情報学専攻 2020年8月実施 専門科目 S-5
Author realball
Description The Fourier spectrum of a continuous-time signal \(f(t)\) is given by \(F(\omega) = \int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt\) ,
where \(j\) denotes the imaginary unit.
Let \(X[k]\) denote discrete Fourier transform of a finite-length discrete signal of length \(N\) , \(x[n] \ (n = 0, \ldots, N-1)\) .
Answer the following questions.
Q.1
Suppose that \(f(t)\) is an even function, that is \(f(x) = f(-x).\) The Fourier spectrum of \(f(t)\) is given by
\[
F(\omega) = 2 \int_0^{\infty} f(t) \boxed{\ (A)\ }dt.
\]
Answer \(\boxed{\ (A)\ }\) . Calculation procedure must also be included in the answer.
Q.2
Let \(x_s[n]\) be the circular shifted version of \(x[n]\) by \(s\) ,
\[
x_s[n] = \begin{cases}
x[N+n-s] &(n<s) \\
x[n-s] &(n \geq s)
\end{cases}
\]
The discrete Fourier transform of \(x_s[n]\) is given by \(\boxed{\ (B)\ }X[k]\) .
Answer \(\boxed{\ (B)\ }\) . Calculation procedure must also be included in the answer.
Q.3
Let \(y[n]\) be a finite-length discrete signal of length \(2N\) ,
\[
y[n] = \begin{cases}
x[n] &(0 \leq n < N) \\
x[2N-1-n] &(N \leq n < 2N).
\end{cases}
\]
The discrete Fourier transform of \(y[n]\) is given by
\[
Y[k] = 2 \boxed{\ (C)\ } \sum_{n=0}^{N-1} x[n] \cos (\boxed{\ (D)\ }).
\]
Answer \(\boxed{\ (C)\ }\) and \(\boxed{\ (D)\ }\) . Calculation procedure must also be included in the answer.
Q.4
The transform of \(x[n]\) ,
\[
X_{DCT}[k] = \alpha_k \sum_{n=0}^{N-1} x[n] \cos (\boxed{\ (D)\ }), \alpha_0 = 1/\sqrt{N}, \alpha_k = \sqrt{2/N}
\]
is known as a discrete cosine tranform (DCT-II), and is used in data compression such as JPEG.
Explain an advantage of the discrete cosine transform compared to the discrete Fourier transform in terms of data compression.
Kai Q.1
\[
\begin{aligned}
\mathcal{F}(\omega)
&=\int_{-\infty}^{\infty} f(t)e^{-j\omega t}dt\\
&= \int_{-\infty}^{\infty}f(t)\left[\cos(\omega t)-j \sin(\omega t)dt \right]\\
&= \int_{-\infty}^{\infty}f(t)\cos(\omega t)dt - \int_{-\infty}^{\infty}f(t)j \sin(\omega t)dt\\
&= \int_{-\infty}^{\infty}f(t)\cos(\omega t)dt\\
&\text{Thus, } (A) = \cos(\omega t)
\end{aligned}
\]
Q.2
\[
\begin{aligned}
X_s[k]&= \sum_{n=0}^{N-1}x_s[n]e^{-j\frac{2\pi}{N}kn}\\
&=\sum_{n=0}^{S-1}x[N+n-s]e^{-j\frac{2\pi}{N}kn}+\sum_{n=s}^{N-1}x[n-s]e^{-j\frac{2\pi}{N}kn}\\
&\text{assume ①:} m=N+n-s;n=m+s-N\\
&\text{assume ②:} m=n-s;n=m+s\\
&=\sum_{m=N-s}^{N-1}x[m]e^{-j\frac{2\pi}{N}k(m+s-N)}+\sum_{m=0}^{N-S-1}x[m]e^{-j\frac{2\pi}{N}k(m+s)}\\
&=e^{-j\frac{2\pi}{N}ks}\sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{N}km}\\
&\text{Thus: } x_s[k]=X[k]e^{-j\frac{2\pi}{N}ks}\\
&\text{Then: } (B)=e^{-j\frac{2\pi}{N}ks}
\end{aligned}
\]
Q.3
\[
\begin{aligned}
Y[k]&= \sum_{n=0}^{2N-1}y[n]e^{-j\frac{2\pi}{N}kn}\\
&=\sum_{n=0}^{N-1}x[n]e^{-j\frac{2\pi}{2N}kn}+\sum_{n=N}^{2N-1}x[2N-1-n]e^{-j\frac{2\pi}{2N}kn}+\sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{2N}k(2N-1-m)}\\
&=\sum_{n=0}^{N-1}x[n]e^{-j\frac{2\pi kn}{2N}}+\sum_{m=0}^{N-1}x[m]e^{-j\frac{2\pi}{2N}k(m+1)}\\
&=e^{\frac{j\pi k}{2N}}\left[\sum_{n=0}^{N-1}x\left[n\right]e^{-j\frac{\pi k}{N}(n+\frac12)}+\sum_{n=0}^{N-1}x\left[m\right]e^{j\frac{\pi k}{N}(n+\frac12)}\right]\\
&=2e^{-j\frac{\pi k}{2N}}\sum_{n=0}^{N-1}x[n]\cos\left[\frac{k\pi}N(n+\frac12)\right]\\
&\text{Thus}: (C)=e^{-j\frac{\pi k}{2N}};(D)=\frac{k\pi}N(n+\frac12)
\end{aligned}
\]
Q.4
Better concentrate energy