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You need to integrate the modulating signal because frequency is the time derivative of phase. Therefore, the typical relationship from introductory calculus holds: $$\phi_i(t) = \int_{-\infty}^{t} \frac{d\phi_i(\tau)}{d\tau} d\tau = \int_{-\infty}^{t} 2\pi f_i(\tau) d\tau$$ For causal signals, the lower limit on the integral changes to zero: $$\phi_i(... 6 What is the advantage of performing the FSK using IQ modulation? You only need one RF oscillator operating at a single frequency, instead of having 2 (or more in the case of M-ary FSK) oscillators operating at separate frequencies for each bit/symbol. Since you only have one oscillator, you don't have to worry about discontinuities in the phase of the ... 5 Both amplitude and frequency modulated radio signals nowadays use quadrature modulation and demodulation as a mean to transfer and receive radio signals. This question is ill-defined – a signal doesn't use quadrature modulation/demodulation, transmitters/receivers do. So, your question is, if we try to "rescue" it, is Do modern FM receivers/... 5 I waited a bit to see if someone else takes the challenge, but since there are no answers yet, I'm providing mine now. 5 Wow, I'm honored by Matt L. doing what I'm often doing: Referring people to GNU Radio. The project actually has a list of recommended literature, but I don't know how well that'd fit you. It's probably still worth looking into. Then, regarding QPSK: Well, it's one of the basic constellations, and you'd probably be best off reading a textbook intro to ... 5 First, the answer to why you do not see a negative voltage is that the output being digital will range from 0 to the maximum digital voltage at the output (+Vs). This will have a DC offset of +Vs/2 which is simply filtered out with a high pass filter (series cap) resulting in a bipolar waveform with negative voltage after the series cap. UPDATE: Based on ... 5 In general, it means several waveforms will coexist in 5G ecosystem. In the context of LTE-based 5G (up to now), several subcarrier spacings (hence CP) coexist. The first and foremost motivation for this is to support not only mobile broadband but also other use-cases such as mMTC and URLLC. Some of benefits of mixed numerology are flexible scheduling and ... 4 How about just using \dfrac{d\phi}{dt} to compute the instantaneous frequency? GNU Octave code (I just made up a sample rate of 2 Msps): pkg load signal; x = csvread('nonlinear_freq_fit.csv'); y = hilbert(x); z = y - mean(real(y)); dphi_dt = diff(unwrap(arg(z))); Fs = 2e6; f_inst = (Fs/2)/pi * dphi_dt; k = [0:length(f_inst)-1]; t = k/Fs; plot(k, f_inst); ... 4 This could happen as discriminator gain is increased with a filter discriminator approach since in many of those approaches the gain would be maximum and linear for small signals only and then the slope of the discriminator slowly goes down coinciding with the results in your plot (such that you no longer get a perfect sine wave out for a sine wave in—- so ... 3 Assuming an FM modulation of a sine wave of frequency f_m. Theoretically, the modulated signal is an infinite sum of cosines at the frequencies f_c + nf_m (n\in \mathbb{Z}) which the amplitudes depend on the coefficients J_n(\beta) (of course in addition to the amplitude of the carrier wave), where \beta is the modulation index and f_c is the ... 3 With GMSK the frequency modulation index is 0.5. With GFSK, the frequency modulation index is larger than 0.5 (the signaling tones are farther apart). The receiving techniques used for GMSK as an FSK work for GFSK. 3 I just skimmed https://github.com/SaucySoliton/PiFmRds/blob/master/src/pi_fm_rds.c#L454 and from what that code looks like, it initializes a clock generator to run at an adjustable clock. Then, it uses the audio amplitudes to modify that clock's frequency in real time. PWM doesn't seem to be involved, aside from the program using the PWM unit to generate ... 3 If t is time, s(t) is the (appropriately scaled) signal, \omega_0 is the angular frequency, and \phi_0 is a phase offset, then phase modulation is$$ t \mapsto \sin(\omega_0 t + \phi_0 + s(t)) $$Different signs for s(t) and/or \phi_0 may be used, depending on conventions and context. Frequency modulation is more complex to write down, because ... 3 There are many ways of demodulating the FSK signal Indeed! What is the advantage of performing the FSK using IQ modulation? Depends. Generally, IQ is the only shape you have your signal in, so using that is not much of a question – a direct downconversion system has IQ signals, and that's what you'll use. Using such a system has a lot of advantages, ... 3 What if they did all use the same carrier? What you're describing is a single-frequency network (SFN). These are in common use for things that are not stupid FM broadcast. The whole truth is that reception from different senders in an SFN just look like heavy multipath propagation, where the sent signal just takes multiple paths of different length to the ... 3 There's simply no inherent relation between carrier frequency and data rate. I assume a classical comms engineering education here, but since that most often introduces the concept of equivalent baseband early, a good explanation would be: This becomes obvious when you think about the complex baseband representation of a passband being the same, ... 3 I simply use unwrapped atan2(IQ(i)) - atan2(IQ(i-1)) to estimate a discrete derivative, then low pass filter to below 15 kHz. Although with a shallow slope, the 1st order approximation to atan() given by Boschen will work just as well. Your noise might be due to not unwrapping the phase delta, or to not low pass filtering after doing the phase ... 3 Frequency by definition is the derivative of phase with respect to time (a change in phase divided by the change in time is frequency). You see this with the radian expression for frequency given by 2\pi f: A frequency of 1 Hz is 1 cycle per second which is 2\pi radians per second. So similarly phase versus time is the integral of frequency versus time. ... 3 That paper (1973) is about more natural, simple and rich audio synthesis using FM (which later became Yamaha's very populer FM synthesis audio chips). In that respect, the carrier frequency is taken very low compared to the modulating tone frequency, unlike in communication applications. Hence the the side lobes cross the zero frequency boundary and are ... 2 Both amplitude and frequency modulated radio signals nowadays use quadrature modulation and demodulation as a mean to transfer and receive radio signals. The use of a quadrature transmitter / receiver is not needed unless you are using a quadrature signal. A simple car radio would likely use something much simpler. Radios that are capable of quadrature ... 2 Checking the book you linked to, this equation refers to FM modulation over a (weakly) non-linear channel, modeled as a memoryless third-order non-linearity (Eq. 82). The third-order non-linearity produces signals at twice and at three times the carrier frequency f_c (Eq. 84). Note that also the frequency deviations of these additional signal components at ... 2 In U.S. broadcast FM, the various subcarriers (pilot tone, stereo L-R, RDS, etc.) are modulated into the upper sidebands (above the 15 kHz L+R/mono audio, but below 100 kHz, for instance 38 kHz DSB for the stereo subcarrier) of the baseband signal before that entire multiplex is used to wideband frequency modulate a single, much higher frequency, FM carrier.... 2 Frequency is the derivative (or 1st difference) of phase. To get an IQ signal from which you can get phase angles, first multiply your wave file samples by a cosine at 3.15kHz and a sinewave at a frequency of 3.15kHz. That will hetrodyne your signal down to baseband IQ. Use atan2 of the IQ array to get an array of angles, take the 1st difference between ... 2 Integration was incorrect. I was integrating it with dx= 1. Doing m(j)*1/Fsoperation in the for loop, I have made dx= 1/Fs now. As can be seen, frequency changes for different amplitudes now. Moreover, resultis a constant, it should be changed toresult2 so that it gives result with respect to time t. 2 Yes. Pulse compression is really just running the returned signal through a pulse matched filter, which is equivalent to cross correlation. If you view it as a pulse matched filter, matched filters are optimal for detection of a signal in AWGN. If you view it as a cross correlation, the output of the correlation will peak when the signal best matches with ... 2 Why not try your hand at working out the differences for yourself? Since you are interested in cosines, take, for example, the signal x(t) = \exp(j(2\pi t + \theta)) which is a complex sinusoid of period 1 and sample it 16 times per second to get 16 samples x[n], n = 0, 1, \dots, 15, where x[n] = x\left(\frac{n}{16}\right) = \exp\left(j\left(\... 2 If the signal frequency is constant and exactly integer periodic in the FFT width, then the transform basis vector for each FFT result bin is the same as a sinusoidal (complex exponential) IQ down-modulator mixer input that starts with a phase of zero at sample 0. Thus the atan2() results for a signal within that FFT result bin should be identical. If you ... 2 Your question is based on a misunderstanding. The RF signal is not directly sampled by the sound card. Instead, the earphone cable just doubles as an antenna for a dedicated FM receiver; the soundcard hasn't got anything to do with FM reception. In fact, it doesn't even "get" radio frequencies – they're filtered out before and fed to the FM receive chain. 2 The \arctan2() function has a jump at \pm\pi which will cause problems when trying to compute a derivative. Since you're using the first difference discrete derivative approximation to compute d\phi/dt, there's a simple way to avoid the jumps. First note that your signal samples can be written polar form:$$s[n] = i[n]+jq[n] = r[n]\cdot e^{\phi[n]}...
If the two nodes run the same clock i would know perfectly where the signal starts from the end of the first chirp, after the fixed period of pause. In this case the signal i try to demodulate would have no phase shifting, correct?! Wrong. The signal travels a distance. The distance it travels inherently means you get a phase shift – a full period ($2\pi$) ...