## Archive for March, 2011

### The former are the happier

March 26, 2011

A delightfully trenchant passage from Fyodor Dostoevsky’s Idiot (part IV, ch. I):

В самом деле, нет ничего досаднее, как быть, например, богатым, порядочной фамилии, приличной наружности, недурно образованным, неглупым, даже добрым, и в то же время не иметь никакого таланта, никакой особенности, никакого даже чудачества, ни одной своей собственной идеи, быть решительно «как и все». Богатство есть, но не Ротшильдово; фамилия честная, но ничем никогда себя не ознаменовавшая; наружность приличная, но очень мало выражающая; образование порядочное, но не знаешь, на что его употребить; ум есть, но без своих идей; сердце есть, но без великодушия, и т. д., и т. д. во всех отношениях. Таких людей на свете чрезвычайное множество и даже гораздо более, чем кажется; они разделяются, как и все люди, на два главные разряда: одни ограниченные, другие «гораздо поумнее». Первые счастливее. Ограниченному «обыкновенному» человеку нет, например, ничего легче, как вообразить себя человеком необыкновенным и оригинальным и усладиться тем без всяких колебаний.

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Here’s Julius Katzer’s translation (1985):

Indeed, nothing is more vexing than to be, for instance, wealthy, of good family, passable in looks, fairly well educated and intelligent, and even kindly, and yet to possess no talent, no outstanding feature or even quirk, not a single idea of one’s own, and to be positively “just like anybody else”. There is wealth, but far less than the Rothschilds possess; the family is an honorable one, but has never won the least distinction; one’s looks are pleasant enough, but express nothing in particular; one’s education is quite sound, but one has no idea of what to direct it towards; one has intelligence but no ideas of one’s own; one has a kind heart but no magnanimity, and so and so forth, on all counts. There is a vast multitude of such people in the world, and even far more than may seem. Like all other people, they fall into two categories: those of limited intelligence, and those that are “far cleverer than most”. The former are the happier. To the “common-place” man of limited intelligence, for instance, nothing is easier than to imagine that he is exceptional and original, and to derive the utmost enjoyment therefrom, without the least hesitation.

And here’s another translation (1913):

There is, indeed, nothing more annoying than to be, for instance, wealthy, of good family, nice-looking, fairly intelligent, and even good-natured, and yet to have no talents, no special faculty, no peculiarity even, not one idea of one’s own, to be precisely “like other people.” To have a fortune, but not the wealth of the Rothschild; to be of an honorable family, but one which has never distinguished itself in any way; to have a decent intelligence, but no ideas of one’s own; to have a good heart, but without any greatness of soul; and so and so on. There is an extraordinary multitude of such people in the world, far more than it appears. They may, like all other people, be divided into two classes: some of limited intelligence; others much cleverer. The first are happier. Nothing is easier for “ordinary” people of limited intelligence than to imagine themselves exceptional and original and revel in that delusion without the slightest misgiving.

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Source:

Федор Михайлович Достоевский, Идиот, 1868.

### Interpolation and energy amplification

March 25, 2011

Consider the following interpolator [1]

where $x_u = (\uparrow L) (x)$ is the upsampling of sequence $x$ by an integer factor of $L$, and where $H(z)$ is the transfer function of an ideal discrete-time LTI lowpass filter with gain $L$ and cutoff frequency $\pi/L$. What is the relation between the energies of signals $x$ and $y$?

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My solution:

It can be easily shown [1] that the interpolator’s input-output relationship is $Y (z) = H(z) X (z^L)$. Hence, evaluating the Z-transforms on the unit circle, we obtain

$Y (e^{j \omega}) = H(e^{j \omega}) X (e^{j \omega L})$.

The energy of the interpolator’s output signal is

$\mathcal{E}_y := \displaystyle \sum_{n \in \mathbb{Z}} | y(n) |^2 = \frac{1}{2 \pi} \int_{-\pi}^{\pi} |Y (e^{j \omega})|^2 d \omega$

where we used Parseval’s theorem for discrete-time signals. Thus,

$\mathcal{E}_y = \displaystyle\frac{1}{2 \pi} \int_{-\pi}^{\pi} |H(e^{j \omega})|^2 |X (e^{j \omega L})|^2 d \omega$.

$H(z)$ is an ideal lowpass filter with the following frequency response

$H(e^{j \omega}) = \begin{cases} L, & |\omega| \leq \pi/L\\ 0, & \pi/ L< |\omega| \leq \pi\end{cases}$

and the following sinc impulse response

$h(n) = \displaystyle\frac{1}{2 \pi} \int_{-\pi}^{\pi} H(e^{j \omega}) e^{j \omega n} d \omega = \displaystyle \frac{\sin(\frac{n \pi}{L})}{\frac{n \pi}{L}} =: \text{sinc} \left(\frac{n \pi}{L}\right)$.

Please do note that this is a non-causal filter with an impulse response of infinite length. We can then write the interpolator’s output signal’s energy as follows

$\mathcal{E}_y = \displaystyle\frac{1}{2 \pi} \int_{-\pi}^{\pi} |H(e^{j \omega})|^2 |X (e^{j \omega L})|^2 d \omega = \displaystyle\frac{L^2}{2 \pi} \int_{-\pi/L}^{\pi/L} |X (e^{j \omega L})|^2 d \omega$.

Performing a change of variable, $\theta = L \omega$, we obtain

$\mathcal{E}_y = \displaystyle\frac{L^2}{2 \pi} \int_{-\pi/L}^{\pi/L} |X (e^{j \omega L})|^2 d \omega = \displaystyle\frac{L}{2 \pi} \int_{-\pi}^{\pi} |X (e^{j \theta})|^2 d \theta = L \mathcal{E}_x$

where

$\mathcal{E}_x := \displaystyle \sum_{n \in \mathbb{Z}} | x(n) |^2 = \frac{1}{2 \pi} \int_{-\pi}^{\pi} |X (e^{j \omega})|^2 d \omega$

is the energy of the input signal. Since $\mathcal{E}_y = L \mathcal{E}_x$, we can conclude that the interpolator amplifies the input signal’s energy by a factor of $L$, which is somewhat intuitive.

We considered the case where signals are bi-infinite sequences and $H(z)$ is a non-causal ideal lowpass filter. It would be interesting to consider the more realistic case where the signals are finite sequences and $H(z)$ is a causal FIR filter.

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References

[1] Alan V. Oppenheim, Ronald W. Schafer, Discrete-Time Signal Processing, 2nd edition, Prentice-Hall, 1999.

### Oddly real

March 12, 2011

Consider a complex-valued continuous-time signal, $x : \mathbb{R} \to \mathbb{C}$, with the property $x(t) = - x^*(-t)$. Show that the even part, $x_e$, and the odd part, $x_o$, of signal $x$ are imaginary and real, respectively. Recall that the even and imaginary parts of a signal are defined as follows [1]

$x_e (t) := \displaystyle\frac{x(t) + x(-t)}{2}, \qquad{} x_o (t) := \displaystyle\frac{x(t) - x(-t)}{2}.$

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My solution:

Since $x$ is a complex-valued signal, we can decompose it into its real and imaginary parts

$x(t) = x_R (t) + j x_I (t)$

where $x_R$ and $x_I$ are real-valued signals. The complex conjugate of $x$ is

$x^*(t) = x_R (t) - j x_I (t)$

and, from $x(t) = - x^*(-t)$, we know that $x(-t) = - x^*(t)$. The even part is, then

$x_e (t) = \displaystyle\frac{x(t) + x(-t)}{2} = \displaystyle\frac{x(t) - x^*(t)}{2} = j x_I (t)$

which is purely imaginary, and the odd part is

$x_o (t) = \displaystyle\frac{x(t) - x(-t)}{2} = \displaystyle\frac{x(t) + x^*(t)}{2} = x_R (t)$

which is real.

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References

[1] Alan V. Oppenheim, Alan S. Willsky, S. Hamid Nawab, Signals & Systems, 2nd edition, Prentice-Hall, 1996.

### Ein blutiger Schleier vor den Augen

March 9, 2011

A passage from Ernst Jünger‘s brutal In Stahlgewittern:

Andererseits muß ein Verteidiger, der dem Angreifer bis auf fünf Schritt seine Geschosse durch den Leib jagt, die Konsequenzen tragen. Der Kämpfer, dem während des Anlaufs ein blutiger Schleier vor den Augen wallte, kann seine Gefühle nicht mehr umstellen. Er will nicht gefangennehmen; er will töten. Er hat jedes Ziel aus den Augen verloren und steht im Banne gewaltiger Urtriebe. Erst, wenn Blut geflossen ist, weichen die Nebel aus seinem Hirn; er sieht sich um wie aus schwerem Traum erwachend. Erst dann ist er wieder moderner Soldat, imstande, eine neue taktische Aufgabe zu lösen.

A possible translation:

The defending force, after driving their bullets into the attacking one at five paces’ distance, must take the consequences. A man cannot change his feelings with a veil of blood before his eyes. He does not want to take prisoners but to kill. He has no scruples left; only the spell of primeval instinct remains. It is not till blood has flowed that the mist gives way to his soul. Only then is he once again a modern soldier able to solve a new tactical task.

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Source:

Ernst Jünger, In Stahlgewittern, Berlin 1922.