haradwaj LS

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  • Mugendorf Hike

    Mugendorf Hike

    The Team

    The university started 1 month back. We have already made some good friends but we haven’t yet met outside the university so far. So we decided to go for a trek. And what better way to plan a trek than give the responsibility to the only German in our class.

    We chose him as the leader and he did an exceptional job as choosing the spot. There were caves, hills and beautiful autumn colors. We decided to go to Muggendorf which is 1 hr from Erlangen.

    We formed the WhatsApp group with 11 people who wanted to join. We ended up with 7 (6 Indian and 1 German) people finally. Some had other priorities and one woke up with a sore back.

    The Group

    Off we go

    The plan was to take the morning RE29 train to Forchheim. From there change to RB22 Ebermannstadt. Finally, a bus would take us to the start of the trek at Mugendorf.

    I was supposed to board the train at 0825 from Erlangen. There were 2 from Nurnberg and they started the journey earlier at 0810. Another friend wanted to join us directly at Forchheim. This guy, we accidently spotted him seated in the train at the Erlangen station so we got him off the train.

    The journey started for some with a small hiccup for some – as the day was the first day of the month, the Deutchland ticket (bus pass) did not renew for some. So, they were not sure if they could travel. One did not join and the other one braved and joined us.

    So the five of us boarded the train but we couldn’t meet the other two inside the train. A short 20 minutes ride brought us to Forchheim. Here we met the other two and we changed the train to RB22.

    Another 20 minutes and we were at the Ebermannstadt station. We changed to the final bus which would take us to the start of the trek.

    And then came of moment of truth — would the mighty PDF ticket rescue the ones whose app wasn’t reflecting the ticket update? Since this was a smaller place the bus driver wanted to check the tickets of everyone. Luckily, he was ok with the PDF and we reached the start of the trail.

    A single coach train

    Cave 1

    The trek was supposed to be a 12 km long trek. The view from the bus stop was already beautiful. The woods had wonderful colors. Knowing that we were going to walk through it filled us with anticipation.

    The hills we climbed

    There were few other people also walking the trail whom we met. We greeted them with the only German words we knew – Hallo!! The woods have so many different trails one could follow that for the most parts we were alone in the woods.

    The trek started with a steep climb through the village. And the we entered the woods. We were walking so slowly pausing every some distance to take a snap. Finally, there was a group of old people. The German friend reminded us that we might get overtaken by the oldies – that hit our pride and we immediately started walking faster.

    In that hurry, we missed a turn and we reached above the cave we were supposed to go into. We then started going down a slippery steep slope for the cave. We noticed that the oldies overtook us.

    Just before we got overtaken

    Nevertheless, the slope was fun on its own. Adventurous and a bit risky. Some already started slipping. But we managed to reach the cave without anyone getting hurt.

    The cave mentioned that it might have bats inside. The entry to the cave was narrow. I pictured myself running out being chased by bats like in Scooby doo.

    The cave was empty and dark. There were some used up candles to the side. I finally saw stalactites and stalagmites for the first time! I knew what they were in theory, but I only realized which was which after our German friend explained it.

    Autumn colors and Cafe

    We continued the hike. We still had 11.3 kms to cover atleast. This first spot was just 700 m but we made it longer because of the missed turn.

    A short distance from there we got the view of the open fields and beautiful autumn colors. The walk was on mostly flat grounds.

    Autumn Colors
    Fields
    A colourful tree
    The long road
    Sunflower

    Berries, Stream and Cafe

    One of our Indian friends asked our German buddy if a berry we found was edible, and he confidently said yes — but I noticed he didn’t eat it himself.

    A little further down the trail, we spotted the same berries again, and this time I saw him pop one into his mouth. I asked if he was sure it was safe, and he grinned, saying he had checked after the first encounter. If the German was eating it, I figured I could too. The berries were sour but was interesting.

    Then we ate the berries as we walked beside a stream. Soon we reached a beautiful cafe. It was a flour mill powered by the stream. The turbine still worked but mostly for aesthetics I guess. It was now a small cafe.

    Cafe
    Stream

    Riesenberg

    At this point we were slightly getting hungry. We ate some fruits and decided to have our lunch at the next view point. It was some 2 kms away.

    We traced our path back along the other bank of the stream.

    Stream enroute Riesenberg

    We reached Riesenberg.This was an interesting structure with use arches. This was originally a cave that caved in, leaving behind the huge arches. This was covered with leaves making the place look more captivating.

    Some passerby offered to click our group photo. This was the second time in the day when people volunteered to click our photo. The first time was the group photo above.

    Collapsed cave
    View from the cave

    The wall had this inscribed on it. Quite ironic since the rock collapsed! Rock is also not permanent afterall.

    “Following the path of the wind, come to the rock — the clouds may move and shift, but the rock stands unchanging through time.”

    Rock Engraving

    Lunch and Cave 2

    We now had to find a good place to have lunch. All the good places were occupied so we climbed up some rocks like a bunch of monkeys. Had our lunch. Hunger indeed is the best appetizer. Everything tasted so delicious.

    Up the rocks

    After the lunch we walked to the next cave. On the way, we met couple of interesting people. An old husband who has collected wild flowers and gifted his wife those. Then a cyclist climbing a very steep climb. To top it all off, a mother was on the hike with her adorable little baby, under one year old, snugly secured in a kangaroo-style baby sling.

    Another steep downhill slide and we made it to the final cave.

    Cave again

    Way back

    The way back was fairly uneventful as we wanted it to be. We were all a bit tired after the 12 kms hike. We were extremely lucky to reach the bus stop at exactly the right time. If we had missed the bus the next bus was after 2 hours.

    I was surprised the whole time to see so many people on hike. I love it that people here are so active and fit. I also met people with cheerful smiles and greetings all along. I also found the locals very welcoming with two of them volunteering to click our snaps even.

    Our German friend was surprised why we clicked so many photos. For us everything was new and for him everything was usual. I secretly wished I could always see things with the same awe as today and I never get used to it!

    Iteniary

    Outbound: Nürnberg Hbf → Muggendorf Forchheimer Str.

    TimeFrom / PlatformTransportDuration
    08:10Nürnberg Hbf, Platform 4 C-FR-Bahn RE 29 → Erfurt Hbf24 min
    08:40Forchheim (Oberfr), Platform 8R-Bahn RB 22 → Ebermannstadt19 min
    09:05EbermannstadtBus 389 → Gößweinstein Freibad12 min

    Return: Muggendorf Forchheimer Str. → Nürnberg Hbf

    TimeFrom / PlatformTransportDuration
    17:41Forchheimer Str., Muggendorf (Oberfr)Bus 389 → Ebermannstadt Bahnhof14 min
    19:04Ebermannstadt, Platform 1R-Bahn RB 22 → Forchheim (Oberfr)19 min
    19:28Forchheim (Oberfr), Platform 4S-Bahn S1 → Neumarkt (Oberpf)40 min
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  • Diwali Party

    Diwali Party

    The Diwali party at my home marked exactly 1 month since I arrived to Germany. During this month, I was lucky to join a Hike, 2 potlucks and 1 very amazing Diwali party.

    My house mate (let’s call her Apurwa for the post) put in a lot of effort in making this a success. She invited around 30 of her friends — and me. Luckily for me, I happen to live here so I was unavoidably invited 😉.

    The preparations had started long before the party with meticulous planning for gifts and food. Every guest received a gift bag thoughtfully filled with lovely card, incense, candles, sweets, candies and dry fruits for each.

    Things really picked up pace 2 days before the party, when some of her friends started coming home to help with the decoration. It was always fun to meet new people as I returned late from the university. It made my evenings interesting to see the house slowly getting decorated.

    Lights
    The entry

    Then came the big day, with a long list of things to be prepare. The party was a potluck but as a host Apurwa wanted to cook Boondi laddu, malai kofta, Black gram, Chaat, and rice.

    She needed help and of course she can’t exactly rely on me. My highest culinary achievement so far has been cooking Sambar in one hr. So, she called on 4 of her friends to help.

    Six of us started this cooking marathon with lots of energy in the afternoon. I wasn’t entirely sure that we would be able to do everything. Amidst the laughs and jokes we somehow managed to cook it all.

    Food! This is not all!!

    I still had one small problem – I hadn’t packed any traditional dress with me. I never expected my first month to be so festive. Fortunately, one of the friends let me borrow the extra kurta which he brought for those who didn’t have one.

    The party started at 6 pm and slowly started gathering momentum. There were plenty of food, drinks, dance and laughs. Guests came from Spain, Columbia, Egypt, Germany, Turkey and many more places. Seeing all of them dressed in Indian wear, dancing to Indian songs and enjoying Indian food was heartwarming.

    The 4 house mates

    As the night went on, some guests started returning home. Then when the late night game of charades started, it pumped the energy back into people.

    The party ended at 3 am – which magically became 2 am as the Daylight saving ended in Germany! Overall, it was a wonderful experience for me. I can’t yet say I made friends – that needs time and trust – but I certainly have made some great acquaintances. Hope to see them again soon.

    I can’t thank everyone enough for such a truly wonderful time.

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  • Equivalent Complex Baseband

    Equivalent Complex Baseband

    Table Of Contents
    1. Introduction
    2. ECB in quadrature form
  • ECB through LTI
  • Summary
  • Reference

    • Introduction

    This is a summary of the topic Equivalent complex representation. In this we discuss why we need such a representation. It is then followed by some more mathematical manipulation which makes it more convenient to generate and transmit signals. Finally we conclude the section with ECB signal and its interaction with Filters.

    Motivation

    When analyzing any digital communication modulation schemes it will be useful we can separate the modulation or the mixing of the carrier from the information itself. This allows us to analyze the modulation independent of the center frequency. This motivation drives us to use the Equivalent complex baseband representation of the transmitted RF signals.
    To develop an intuition about this process it is easy to think of the transmitted RF signal as just the up-converted baseband signal. So, ideally a down-conversion should bring us back to the baseband. This means that if we can write the baseband signal for any RF signal then we can neglect the carrier frequency and study the modulation at the baseband. In this case the baseband signal is the ECB and of the transmitted RF signal.
    However, the details reveal a subtle complication. Now let us consider the reverse situation and ask this question. What if the RF signal was not generated in baseband and up-converted? Instead if RF frequency was generated directly, can we now say for sure that the down-converted signal has the same information as the RF signal? This is the reason behind the detailed analysis that follows.

    Mathematical Reasoning

    ECB is an equivalent representation of the signal which gets transmitted (RF signal) . The important thing to notice is that the only real signals can be transmitted physically. This is an important fact around which the entire ECB representation works.

    Now for real signals we have

    $$
    X(f) = X^*(-f)
    $$

    [!Note] Proof: \(X(f) = X^*(-f)\)

    $$
    \begin{aligned}
    \mathcal{F}(x(t)) = X(f) &= \int_{-\infty}^{\infty}x(t)e^{-j\omega t} dt \\
    \mathcal{F}(x^*(t)) &= \int_{-\infty}^{\infty}x^*(t)e^{-j\omega t} dt \\
    &= \left(\int_{-\infty}^{\infty}x(t)e^{j\omega t}\right)^* dt \\
    &= X^*(-f)
    \end{aligned}
    $$

    Since x(t) is real =>

    $$
    x(t) = x^*(t)
    $$

    Thus,

    $$
    X(f) = X^*(-f)
    $$

    The above statement means that the negative frequency holds the same information as the positive. Hence we can get rid of one side. We shall be using this fact in the next steps.

    Analytic Signals from RF

    Before we get to the baseband representation (ECB) we encounter what is called the analytic representation. The analytic signal is the RF signal without the negative part of the frequency spectrum. This still has the same information as the RF signal as seen in the previous section.
    Already at this point the signal seizes being a real signal and it can’t be transmitted directly anymore . The analytic signal is represented by \(x^+(t)\) and its Fourier transform \(X^+(f)\). In this section we get to the time and frequency domain representation of analytic signals.
    The idea is straight forward. We just need to keep the positive side and subtract the negative. Mathematically it can be achieved using the signum function (also called sign function) defined as

    $$
    \operatorname{sgn}(x) =
    \begin{cases}
    -1, & x < 0 \\ 0, & x = 0 \\ 1, & x > 0
    \end{cases}
    $$


    $$
    X^+(f) = c[1+sgn(f)]X_{RF}(f)
    $$

    we use the constant c since we would like to keep the energy of both the signals same we can compute c. c can be computed as follows

    $$
    \begin{aligned}
    E_{X_{RF}} &= E_{X^+}\\
    E_{X_{RF}} &= \int_{-\infty}^{\infty}\left|c[1+sgn(f)]X_{RF}(f) \right|^2\\
    &= \int_{0}^{\infty}\left|c2X_{RF}(f) \right|^2\\\
    &= 2c^2E_{X_{RF}}\\
    c &= \frac{1}{\sqrt{2}}
    \end{aligned}
    $$

    Taking the inverse Fourier transform we get

    $$
    \begin{aligned}
    x^+(t) &= \frac{1}{\sqrt{2}}[x_{RF}(t) + x_{RF}(t)*\mathcal{F}^{-1}(x_{RF}(t))]\\
    &= \frac{1}{\sqrt{2}}[x_{RF}(t) + j(x_{RF}(t)*\frac{1}{\pi t})] \quad \text{[Since $\mathcal{F}^{-1}(x_{RF}(t)) = \frac{j}{\pi t}$]}\\
    &=\frac{1}{\sqrt{2}}[x_{RF}(t) + j\mathcal{H}(x_{RF}(t))]
    \end{aligned}
    $$

    where
    \(\mathcal{H}(x(t)) = x_{RF}(t)*\frac{1}{\pi t}\) is called the Hilbert transform

    [!note] Proof: \(\mathcal{F^{-1}}(sgn(f)) = \frac{j}{\pi t}\)

    $$
    \mathcal{F^{-1}}(sgn(f)) = \int_{-\infty}^{0}e^{j2 \pi f t}df + \int_{0}^{\infty}e^{j2 \pi f t}df
    $$

    This clearly is not integrable as the two integrands are sinusoids. Hence, we need a trick.
    We will define 4

    $$
    \begin{aligned}
    sgn(x) &= \lim_{a \to 0} e^{-a|f|} sgn(f) \\
    &= \lim_{a \to 0}[e^{-af}u(f) – e^{af}u(-f)] \\
    \end{aligned}
    $$

    Now taking the inverse Fourier transform

    $$
    \begin{aligned}
    \mathcal{F^{-1}}(X(f)) &= \int_{-\infty}^{\infty}X(f)e^{j2 \pi f t} df \\
    &= \lim_{a \to 0}\left[\int_{0}^{\infty}e^{-af}e^{j2 \pi f t}df – \int_{-\infty}^{0}e^{af}e^{j2 \pi f t}df \right]\\
    &= \lim_{a \to 0}\left[\int_{0}^{\infty}e^{-(a – j2 \pi t)f}df – \int_{-\infty}^{0}e^{(a + j2 \pi t)f}df \right]\\
    &= \lim_{a \to 0} \left[ \frac{e^{-(a – j2 \pi t)f}}{-(a – j2 \pi t)} \right]_{0}^{\infty} – \lim_{a \to 0} \left[ \frac{e^{(a + j2 \pi t)f}}{a + j2 \pi t} \right]_{-\infty}^{0} \\
    &= – \frac{1}{j2\pi t} – \frac{1}{j 2 \pi t}\\
    &= \frac{j}{\pi t}
    \end{aligned}
    $$

    Transformation: RF to ECB

    Now that we have the analytic signal, ie. the signal where the negative frequencies are 0 it is a matter of simple downshift to get the baseband.
    Thus we have

    $$
    X(f) = X^+(f+f_0)
    $$

    or equivalently we have in time domain
    $$
    \begin{aligned}
    x(t) &= x^+(t)e^{-j2\pi f_0t}\
    &= \frac{1}{\sqrt{2}}[x_{RF}(t) + j\mathcal{H}(x_{RF}(t))]e^{-j2\pi f_0t}
    \end{aligned}
    $$

    Finally we have the important relation that converts RF to ECB

    $$
    x(t) = \frac{1}{\sqrt{2}}[x_{RF}(t) + j\mathcal{H}(x_{RF}(t))]e^{-j2\pi f_0t}
    $$

    Transformation: ECB to RF


    This is the reverse transformation of what we did so far. We would like to get the RF signal from the ECB. This part is relatively straight forward. We just need to take the baseband signal and up-convert and keep only the real part. The real signal is the original RF signal we started with

    $$
    x_{RF}(t) = \sqrt{2}Re(x(t)e^{j2\pi f_0t}) = \frac{x(t)e^{j2\pi f_0t} + x^*(t)e^{-j2\pi f_0t}}{\sqrt{2}}
    $$
    or the Fourier transform
    $$
    X_{RF}(f) = \frac{1}{\sqrt{2}}(X(f-f_0) + X(-(f+f_0))
    $$


    *The above expression takes the ECB and shifts by up by \(f_0\). Also it makes another copy which reflects around 0 and shifts down by \(f_0\)*

    ECB in quadrature form

    Motivation

    We saw in [[Equivalent Complex Baseband]] that the RF signal can be represented by ECB. ECB is a complex signal. So the question remains on how to generate these signals in the real world. This section provides a practical way of producing these signals and using it.

    ECB Quadrature to RF

    For practical use we would like to generate these ECB signals at baseband and have a way to modulate this to the RF frequency. We start with the describing expanding the ECB into real and imaginary part. And then we proceed by substituting these expressions to get the relevant RF.

    We can split the ECB signal into real and imaginary part and call it in phase and quadrature components. The reason for the naming becomes apparent when the derivation ends with multiplication of cos and sin waves.

    $$
    x(t) = x_I(t) + jx_Q(t)
    $$

    Then we know to get RF frequency it needs to be multiplied by the center frequency and considering only the real part

    $$
    \begin{aligned}
    x_{RF}(t) &= \sqrt{2}Re[x(t)e^{j2\pi f_0t}] \\
    &= \sqrt{2}Re((x_I(t) + jx_Q(t))(\cos{2\pi f_0t} + j \sin{2\pi f_0t})\\
    &= \sqrt{2}(x_I(t)\cos{2\pi f_0t} – x_Q(t) \sin{2\pi f_0t})
    \end{aligned}
    $$
    $$
    \boxed{x_{RF}(t)= \sqrt{2}(x_I(t)\cos{2\pi f_0t} – x_Q(t) \sin{2\pi f_0t})}
    $$

    RF to ECB quadrature

    The section derives the general conversion first. It is then followed by the more practical conversion as well.

    In order to obtain the ECB from RF we have to: get the analytic function, down-convert and scale

    $$
    \begin{aligned}
    x(t) &= \frac{1}{\sqrt{2}}x^+(t) e^{-j2\pi f_0t} \\
    &= \frac{1}{\sqrt{2}} (x_{RF}(t)+ j\mathcal{H}(x_{RF}(t))(\cos {2\pi f_0t} – j\sin{2\pi f_0t})\\
    &= \frac{1}{\sqrt{2}}(x_{RF}(t)\cos{2\pi f_0t}+\mathcal{H}(x_{RF}(t)\sin{2\pi f_0t}) + j (\mathcal{H}(x_{RF}(t)\cos{2\pi f_0t} – x_{RF}(t)\sin{2\pi f_0t})
    \end{aligned}
    $$

    Thus we have

    $$
    \begin{align}
    \boxed{x_I(t) = \frac{1}{\sqrt{2}}(x_{RF}(t)\cos{2\pi f_0t}+\mathcal{H}(x_{RF}(t)\sin{2\pi f_0t})}\\
    \boxed{x_Q(t) =\frac{1}{\sqrt{2}}(\mathcal{H}(x_{RF}(t)\cos{2\pi f_0t} – x_{RF}(t)\sin{2\pi f_0t})}
    \end{align}
    $$

    The above expressions can convert any RF frequency down to its ECB in quadrature form. This expression is particularly complex due to the presence of Hilbert transformer and so many terms. Firstly a Hilbert transformer which spans the entire frequency range is not practical and these many terms complicate the process.

    Intuition behind the simpler version

    This can be simplified if we make an assumption that the our signal of interest is usually band limited. This assumption simplifies the case as we don’t have to bother about the entire spectrum from 0 to infinity. In other words, the spectrum is 0 everywhere other than the band of interest. Under this assumption one can avoid the Hilbert transform but instead use a low pass filter.

    Mathematical proof

    From the basic definition of ECB we have

    $$
    \begin{align}
    X(f) = \frac{1}{\sqrt{2}}(1 + sgn(f+f_0))X_{RF}(f+f_0)
    \end{align}
    $$

    The signum function is 0 below \(-f_0\) and 1 after that. But since the signal is band-limited this implies that the signal after down-conversion resides only between \(-f_0 \leftrightarrow f_0\) . Thus \((1 + sgn(f+f_0)\) can be replaced by a rectangle centered at 0 with a width of \(2f_0\). This in frequency domain is a low pass filter.
    Thus we have

    $$
    \begin{align}
    X(f) &= \sqrt{2}.rect\left( \frac{f}{2f_0} \right)X_{RF}(f+f_0) \quad \text{[Since $(1 + sgn(f+f_0)$ scales to 2]}\\
    \end{align}
    $$

    Taking inverse Fourier transform

    $$
    \begin{align}
    x(t) &= \sqrt{2}h_{LP}(t)*(x_{RF}(t)e^{-j2\pi f_0t})\\
    &= \sqrt{2}h_{LP}(t)*(x_{RF}(t)(\cos{(-2\pi f_0t)} + \sin{(-2\pi f_0t)})\\
    \end{align}
    $$
    $$
    \begin{align}
    \boxed{x_I(t) = \sqrt{2}h_{LP}(t)*(x_{RF}(t)(\cos{(2\pi f_0t)}}\\
    \boxed{x_Q(t) = -\sqrt{2}h_{LP}(t)*(x_{RF}(t)(\sin{(2\pi f_0t)}}\\
    \end{align}
    $$

    This is the same as what was described in the intuition section.

    Analysis on synchronization error

    We now have expressions to go from ECB to RF and back. ECB to RF is done at the transmitter and RF to ECB is done at the receiver. Often there will be an offset in the frequency and phase of the oscillators at the TX and RX. This section we shall see how the received signal gets impacted.

    Let us assume that ECB signal at the transmitter be

    $$
    s(t) = s_I(t) + j s_Q(t)
    $$

    Thus we get the transmitted RF as

    $$
    s_{RF}(t) = \sqrt{2}Re(s(t)e^{j2\pi f_0t})
    $$

    The received signal now is given by

    $$
    \begin{align}
    r(t) &= \sqrt{2}h_{LP}(t) * (s_{RF}(t)e^{-j2\pi (f_0 + \Delta f)t + \phi })\\
    &= 2 h_{LP}(t)*[(s_I(t)\cos({2\pi f_0 t}) – s_Q(t)sin({2\pi f_0 t}))(\cos({2\pi (f_0 + \Delta f)t + \phi })-j\sin({2\pi (f_0 + \Delta f)t + \phi }))]\\
    &=s_I(t)(\cos(2\pi\Delta ft+\phi)+j \sin (2\pi\Delta ft+\phi)) + s_Q(t)(\cos(2\pi\Delta ft+\phi)+j \sin (2\pi\Delta ft+\phi))\\
    &= (s_I(t) + js_Q(t))e^{-j2\pi \Delta ft + \phi }\\
    &= s(t)e^{-j2\pi \Delta ft + \phi }
    \end{align}
    $$

    Thus we have that the received signal is multiplied by a complex exponential whose frequency is the difference in frequency and phase is the difference in phase

    $$
    \boxed{r(t) = s(t)e^{-j2\pi \Delta ft + \phi }}
    $$

    ECB through LTI

    Motivation

    Any digital system would make use of filters. Using the ECB representation we are able to bring the RF frequency down to the base-band. Now we would like to determine the equivalent filter that should operate in base-band frequencies so that we get the same response as the filter at the HF frequency.
    We would like to initially know if such a system exists. If it does then we would like to know how to build such a system.

    Proof of existence

    We start with a filter in RF frequency with the following setup at the RF frequency

    $$
    Y_{RF}(f) = X_{RF}(f)H_{RF}(f)
    $$

    Let us assume that we have a ECB works the same way for H(f) as it works for X(f) and Y(f) we should get

    $$
    Y(f) = H(f)X(f)
    $$

    where

    $$
    \begin{align}
    X_{RF}(f) &= \frac{1}{\sqrt{2}}(X(f-f_0) + X^*(-(f+f_0)))\\
    Y_{RF}(f) &= \frac{1}{\sqrt{2}}(Y(f-f_0) + Y^*(-(f+f_0)))\\
    H_{RF}(f) &= c_h(H(f-f_0) + H^*(-(f+f_0)))\\
    \end{align}
    $$

    If this was true, then we should be able to start with the RHS and get the LHS.

    $$
    \begin{align}
    X_{RF}(f)H_{RF}(f) &= \frac{c_h}{\sqrt{2}}(X(f-f_0) + X^*(-(f+f_0)))(H(f-f_0) + H^*(-(f+f_0)))\\
    &= \frac{c_h}{\sqrt{2}}(X(f-f_0)H(f-f_0) + X^*(-(f+f_0))H^*(-(f+f_0)) \\
    & +X(f-f_0)H^*(-(f+f_0)) + X^*(-(f+f_0))H(f-f_0))\\
    &= \frac{c_h}{\sqrt{2}}(X(f-f_0)H(f-f_0) + X^*(-(f+f_0))H^*(-(f+f_0)) \\
    &= \frac{c_h}{\sqrt{2}}(Y(f-f_0)+ Y^*(-(f+f_0)))
    \end{align}
    $$

    this will be equal to \(Y_{RF}(f)\) if \(c_h = 1\)

    So we have,

    $$
    \begin{align}
    H_{RF}(f) &= (H(f-f_0) + H^*(-(f+f_0)))\\
    \end{align}
    $$

    [!note]
    There is a difference in scaling between the H and X.

    The difference in scaling makes the ECB for transfer function different by a scaling factor.
    RF to ECB:

    $$
    \boxed{h(t)= \frac{1}{2}[h_{RF}(t) + j\mathcal{H}(h_{RF}(t))]e^{-j2\pi f_0t}\}}
    $$

    ECB to RF:

    $$
    \boxed{
    h_{RF}(t) = 2Re(h(t)e^{j2\pi f_0t}) = h(t)e^{j2\pi f_0t} + h^*(t)e^{-j2\pi f_0t}
    }
    $$

    ECB filter action

    The ECB representation of both the input and the filter response in complex in the ECB. Hence they can be expanded to see how it can be realized

    $$
    \begin{align}
    h(t) &= h_I(t) + jh_Q(t)\\
    x(t) &= x_I(t) + jx_Q(t)\\
    y(t) &= h(t)*x(t)\\
    &= ( h_I(t) + jh_Q(t))*(x_I(t) + jx_Q(t))\\
    &= (x_I(t)*h_I(t)-x_Q(t)*h_Q(t)) + j(x_Q(t)*h_I(t) + x_I(t)*h_Q(t))
    \end{align}
    $$

    Simplification

    There are the following steps to get the ECB from RF

    1. Get the analytic signal
    2. Translate down
    3. scale
      Since the input signal is already analytic for a ECB system, we can relax the condition on the transfer function. This means for the transfer function we can skip step 1 and the end result would still be the same filter provided that the input is ECB.

    They an alternate simpler system

    $$
    \tilde{H}(f) = H_{RF}(f)e^{-j2\pi f_0t}
    $$

    will perform the same way as H(f).

    Summary

    We see that ECB is a convenient representation of the signals in baseband. This helps in studying the modulation independent of the carrier frequency. We also see how the signals can be transformed between these two states. Finally, we also deduce the effects of mismatch frequency and phase between the transmitter and the receiver on the ECB representation.

    Reference

    1. Fischer, R. F. H., & Huber, J. B. (2024). Digital Communications: A Foundational Approach. Cambridge: Cambridge University Press.
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  • Masters! Get set! Go!

    Masters! Get set! Go!

    It has been nearly two weeks since the university has started. I am adjusting to the new normal. It is getting tough to find free time, but today I have some.

    I have started to get a feel for the way education works here. I experience plenty of differences, from the way the course is structured to the way the course is taught. Also, the tests are going to be different.

    So let’s start.

    Table Of Contents
    1. Course Structure
    2. Tenure
    3. Registering for courses
    4. Registration for Exams
    5. Attendance & Assignments
    6. Reference Material
    7. Knocking on tables
    8. Semester Exams
    9. Reflections on the first two weeks

    Course Structure

    The course structure offers a lot of flexibility in the choice of subjects. This is both good and bad. The availability of so many subjects is good. But with it also bring the stress of making decision.

    The university has strong push on the academic independence and hence encourages students to make choices according to their desires. There is also a study advisor in case one is not able to decide.

    So in my program, for instance, I would need 120 credits in order to complete the Master’s Program. Out of these 30 credits are mandatory subjects. This leaves 90 credits which comprises labs, seminar, thesis and electives which can be chosen from a vast catalog.

    Tenure

    Also, what is flexible is the master’s tenure. Though one can complete the master’s in four semesters but one need not. It was mentioned that the more realistic schedules span five semesters.

    It is also possible that one might study more subjects during 5 semesters or 6 and get more than the necessary 120 credits.

    This means the study plans can be completely different for each individual. So, the batch that starts off together may not be even meeting regularly.

    In my own class I already see few people inclined towards audio processing, few others in communication and some towards Signal processing.

    Registering for courses

    The next step after making the plan is to join the courses. This process is called Registering. Registrations turn out to be crucial for the lab courses where the seats are limited. So, is the case with some of the hot subjects.

    Usually ,more than 1 choice is needed for the lab course registration. This way if the first choice is not available, then the alternative will be provided.

    Registration also helps the professor communicate easily with the students. They can then share the lecture notes or other resources. Also, the registered students will get informed if the classes get canceled or rescheduled.

    Registration for Exams

    Now, registration for a course does not imply that one can answer the exam. There seems to be a separate registration for the exams.

    People can choose to attend the classes but not answer the exam. Unattempted exams don’t count as failture. One would fail only if they attempt it.

    Talking about failure, one gets 4 attempts to pass a subject. If the person fails a reattempt is possible only in the next semester. This is unlike back home where the reattempt happens immediately.

    After 4 attempts, the student is ex-matriculated (expelled from the University). Since the mandatory subjects can’t be changed they need to be passed within 4 attempts. Whereas, for the electives one can change the elective if he struggles.

    Attendance & Assignments

    The lectures here are 1.5 hrs each. The best part here so far is that attendance is not mandatory. Though it is highly recommended. Similarly, there are no assignments over the course of the lecture period.

    Good! Yes, but it also would mean that the individual has to be disciplined to study regularly. Otherwise, it is easy to not study for the entire semester and be faced with an insurmountable exam in the end.

    Reference Material

    How would one know what to study if one doesn’t attend classes? The reference material…

    The lecture notes also referred to as script by the professors are available in its entirety. The things taught in the class closely follow the script.

    It is possible to read the script (that’s like 300-400 slides) at home and still do well in the test. Also, any other reference works equally good as long as all the topics are studied.

    Knocking on tables

    The end of lectures have a surprise in it. German’s knock on the table as a sign of appreciation to the lecturer. This is similar to clapping after the end of a presentation. It also serves to wake up sleepy students!

    Semester Exams

    I can’t comment about it just yet. But there are already a few peculiarities.

    First, the exam might be written or oral. It depends on the professor. If it is written then there is often an option to answer the paper in German.

    Secondly, we can carry cheat sheets! I was surprised when I heard that. Most subjects allow one or two A4 sheets written with whatever. But it needs to be hand written. Some subjects also allow carrying any amount of material like even books. But, the rule of thumb that I understand is that the more flexible the subject is regarding carrying cheat sheets, the more difficult the exam is.

    Reflections on the first two weeks

    Overall, the past weeks has been fascinating. I find myself slowly settling into a rhythm. I am now able to manage feeding myself and attending lectures. This was my primary concern before the classes started.

    The new worry that is replacing the old is about how to manage the pace of these subjects that I have taken. But eventaully this will settle too.

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  • Photo of the day

    A wall full of climbers

  • Follow this Blog

    Follow this Blog

    I have been asked in the previous week on how to follow this blog. If you have found it useful there are two ways you can follow it. You can follow using RSS reader or using any social media that supports Fediverse.

    I thought about enabling email notifications but I now decided against it. For what I know one will surely start disliking the mails and it is bound to get ignored. I know because I do the same as well.

    Table Of Contents
    1. RSS reader (Preferred)
    2. Fediverse

    RSS reader (Preferred)

    RSS readers are convenient ways to follow any website. To know more about it you can read here: RSS – Consume internet consciously.

    To cut the long story short you can simply follow the steps below.

    1. Install an RSS reader. Some good ones are
    2. Go to the bottom of this webpage and click on the RSS icon. (A white dot in the bottom-left corner and two curved white line)
    3. Copy the URL and paste it in the RSS reader of your choice.

    Fediverse

    Fediverse is a distributed social media for people who would like to keep their privacy. You can read more about it here. There are several apps that are part of Fediverse like Mastadon, Peertube etc.

    In this blogs we will talk about using Threads as this is the most well know app. Important to notice is that Fediverse is in Beta in Threads. One major issue that I observe is that it only displays short posts. Hence I will highly recommend using the above RSS reader method currently.

    Firstly, to use Threads in Fediverse it needs to be enabled. Then you can look for the blog and follow it just like any other account.

    Enabling Fediverse

    You can find the detailed instructions on the official site. I am just copying it here for convenience.

    1. Click Menu in the bottom left, then click Settings.
    2. Click Account at the top, then click Fediverse sharing.
    3. Follow the onscreen prompts to learn more about sharing to the fediverse.
    4. Once you’ve reviewed the information, click Turn on sharing.

    Following the blog

    1. Go to the bottom of the page and click on Fediverse button.
    2. Copy the profile address which appears there (@blogs).
    3. Search for this address in your Fediverse app and follow.
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  • Kolkar

    Kolkar

    Ingredients

    1. Aunty powder: 2tbsp
    2. Oil: 2tbsp
    3. Garlic: 15 pieces
    4. Salt: to taste
    5. Tamarind juice: to taste
    6. Sugar: 0.5 tsp
    7. Mustard: 0.25 tsp
    8. Jeera: 0.5 tsp
    9. Pepper: 8 – 10
    10. Water: 0.5 l

    Procedure

    1. Mix aunty powder, salt, sugar, tamarind water (to taste) and water (consistency should not be too thin).
    2. Heat oil in pressure cooker and then add mustard, jeera, pepper.
    3. Fry garlic till it becomes reddish.
    4. Add this to the mix made earlier.
    5. Cook for 2 whistle and leave it to simmer for 5 min
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