19

The answer is the same to the question: "Why do we need computers to process data when we have paper and pencil?" DTFT as well as the continuous-time Fourier Transform is a theoretical tool for infinitely long hypothetical signals. the DFT is to observe the spectrum of actual data that is finite in size.


12

TL, DR: world pervasive algorithms (FFT-related)! The continuous Fourier transform, the Discrete-time Fourier transform (DTFT) and the Discrete Fourier transform (DFT) share conceptually similar traits (regarding energy, convolution, shift, scale, etc.) The DFT unveiled a very scalable and fast algorithm to put those concepts into practice: the FFT. It is ...


8

When you quantize a signal, you introduce and error which can be defined as $$q[n] = x_q[n]-x[n]$$ where $q[n]$ is the quantization error, $x[n]$ the original signal, and $x_q[n]$ of the quantized signal. The maximum quantization error is simply $max(\left | q \right |)$, the absolute maximum of this error function. Dx in this definition seems to be the ...


6

The sampling theorem states that $f_\mathrm{S} \geq 2f_\mathrm{max}$, where $f_\mathrm{S}$ and $f_\mathrm{max}$ are the sampling and maximum signal freuqency, respectively. But there's an additional condition: The equal sign only holds if the signal spectrum does not contain a dirac impulse at $f_\mathrm{S}/2$ which is clearly the case in your example. ...


6

The sample rate needs to be GREATER than (NOT just equal to) twice the highest non-zero frequency content of the signal being sampled. Just a little bit greater might work, but the closer the sample rate is to twice the signal frequency, the longer in time you may need to sample to raise the signal above the noise and complex conjugate image in a DFT/FFT ...


6

To add here are two diagrams showing common transceiver architectures: (1) a super-heterodyne where the down-conversion is done first to an IF frequency and then to baseband and (2) a zero-IF receiver where RF is translated directly to baseband. Note in both architectures it is arbitrary (technical / technology choice) where the ADC/DAC boundary is as either ...


6

You're spot on: whereas other frequencies are typically subject to noise that is somewhat benign shaped and result of random processes, DC is usually affected by things like a DC offset. Physically, that happens pretty easily: Say, you've got an ADC with 14 bits of effective number of bits, and it has a differential sensing range of 0 V to 2 V. Then, but 1/...


5

Depends on assumptions you are willing to make and what type of signals are you trying to sample, but in theory I think that sampling rate equal to the Planck time would be a gold standard for anything... This translates to sampling frequency of $1.855 \times 10 ^ {43} \mathtt{Hz}$ ($18.55$ tredecillion hertz). Personally I believe that machines will never ...


5

I have a doubt about (Edit: this was later removed from the question): The distribution of these AM and PM noise components can be reasonably assumed to be uniform as long as the input signal is uncorrelated to the sampling clock Consider the signal: $$\operatorname{signal}(t) = \cos(t) + j\sin(t)$$ and its quantization: $$\operatorname{quantized\_signal}...


5

Sampling is the process of making the x-axis (time) discrete and quantization is the process of making the y-axis (magnitude) discrete. You can sample without quantization (such as done with an analog sample and hold circuit). Quantization is introduced through rounding or truncation when the sampled analog signal is mapped to a digital representation. ...


5

can we have re-configurable analog filters? Yes. The knob you turn on your grandma's kitchen or living room radio changes the tuning of an oscillator by changing the capacitance of a component. Any stable oscillator is essentially a filter for its resonant frequency. Also, plenty of other examples: Tunable RC filters with adjustable R; mechanically tunable ...


5

OP clarified that the question in the comments as follows: If we ignore any modulation for now and assume that we are receiving pure tones plus the band limited noise and we try to improve the SNR in post processing how much improvement can we expect by oversampling and is there a limit to it? My original question was about this aspect. First consider the ...


4

No! The ADC (delta sigma or not) can not reduce the uncertainty in the input. It sounds to me that your friend has not made up a real signal flow diagram and then formed equations. The answer to your second question is affirmative. In fact it's generically true that putting as much gain, without introducing extra, in the front end is a good idea. But......


4

Complex sampling does not "break" Nyquist. IQ quadrature sampling produces twice as many bits per second of information (at the same sample rate for real or complex samples), and the 90 degree phase offset between the I and Q channel in those bits provides extra information about the spectrum. One typical example to demonstrate aliasing is that ...


4

First, use a timer and an ISR to get accurate timing (don't forget to configure the NVIC so that this timer interrupt takes over any other ISR that would be running). Only this will ensure a consistant sample rate. Little variations in timing would create noticeable degradations of audio quality. In particular, in your current example, unless the "filters" ...


4

There are two aspects to how this works. First, since the signal is oversampled there is a great deal of correlation between samples that we can take advantage of via the low-pass filter. The noise, on the other hand, has no correlation (assuming it is white noise), and thus will often destructively interfere with itself. Your question seems to be more ...


4

I'd like to point out Heisenberg Uncertainty principle, based on which theoretical achievable precision is limited. It states that one can not measure two complementary qualities (e.t. here time and charge) concurrently and there is a trade off between amount of precision you can get from one or another. In ADCs, for example theoretical limit for resolution ...


4

First of all, because it's easy to build a 1-bit ADC. It's a comparator. It's literally the easiest ADC you can build. The $\Delta\Sigma$ ADC was invented (or, rather, published) in 1962ยน ! The 2-bit ADC is more than twice as complex as that, you need some window decision: so if you have the choice of making your 1-bit ADC run faster or building a somewhat ...


4

The sampling is indeed analogous to mixing as to my understanding. In the sampling process, we multiply the time domain signal with an impulse train - the impulses in time are represented as impulses in frequency at integer multiples of the sampling rate. So instead of one or two (for a real sine wave) impulses in frequency, we have an infinite number but ...


4

First you will need to determine the number of quantization levels. I am going to assume a power of two for digital convenience's sake. nbits = 8 % 256 qantization levels qLevels = 2^nbits The next step will be to scale your signal to have the same magnitude as your number of bits. signalMin = -1 signalMax = 1 scalingFactor = (signalMax-...


4

I2S audio samples are signed two's complement. Just add $2^{N-1}$, where $N$ is the number of bits, to the result, and binary and by $2^N-1$, to get the range to $0\ldots2^{N}-1$, which I think you used to get from the built-in analog-to-digital converter (ADC). Do this both to the data you receive and the data you transmit using I2S. You can optimize the ...


4

My input signal is a dc signal (sensor output) and im getting kind of a headache to understand why oversampling can increase the resolution of dc signals. The ADC puts out integers. So, let $x$ be the integer that would come out of the ADC (100.3, say, or -333.3). Now let $y = \lfloor r \rfloor$ be the quantization operation, where you get straight ...


4

There really is no practical difference between the two. The important thing to understand about aliasing is not the exact definition of the word, but rather the concept: when two frequency bands alias and there is a signal in one of these bands, after sampling you can't tell which band the signal came from. In a sense, by the time you're asking what is or ...


3

Cross correlation should work. I think the problem is the waveform that you are using. A square wave has bad auto-correlation properties. If it is a periodic square wave it will have multiple peaks. It sounds like you are just using a single pulse which is better, but it will still have a gradual roll-off which is a problem. Instead, use a Barker code, ...


3

No, and the reason is not so much a question of how fast one can sample a continuous-time signal (as the accepted answer and another one says) but rather the impossibility of representing a real number with perfect accuracy via a quantized representation of the real number (as noted in the answer by Marcus Muller). At best, even if we assume an infinite ...


3

Is it theoretically possible to perfectly quantize a continuous signal? No. A quantization has an information content obviously countable as bits. Now, if you have a continuously distributed 1D random variable $X$, then the event that any of these real numbers $x$ occurred is unbounded ("infinite"): $$I(x) = -\log_2\left(P(X=x)\right)$$ So, for ...


3

By definition, band-limited signals in the sense of the sampling theorem have finite energy. Sine waves are periodic and thus have infinite energy. So any dirac pulse in the Fourier transform is not permissible. To be more precise, the sampling theorems only applies to signals that can be represented as $$x(t)=\int_{-f_s/2}^{f_s/2} X(f)\,e^{2\pi i\,ft}\,df$...


3

again, even with the Addendum, i think Lutz's answer misses the point. the point is (quoting Wikipedia): To illustrate the necessity of $f_s \ > \ 2B$, consider the family of sinusoids (depicted in Fig. 8) ) generated by different values of $\theta$ in this formula: $$x(t) = \frac{\cos(2 \pi B t + \theta )}{\cos(\theta )}\ = \ \cos(2 \pi B t) - \sin(2 ...


3

Signals can be classified in many ways, and one of them is according to their nature such as being deterministic or random (stochastic). When a signal is deterministic, its spectral content is given by the Fourier transform provided that the signal is absolutely (or square o.w.) integrable; i.e., its Fourier transform exists. On the other hand when an ...


3

I don't quite understand why you feel "harmonics" are relevant to this discussion. A 5 Mhz has sine wave has no harmonics. If the signal has harmonics, it is not a sine wave any more but a different signal (rectangular, triangle, etc.) For ANY signal: you need to determine the highest frequency that's in the signal and then chose the Nyquist frequency to be ...


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