Frequency aliasing (or folding)
In the following Python script it is shown how the aliasing effect (having signals whose frequency is beyond half of the sampling frequency, i.e., …
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DFT as a change of basis
DFT can be thought as a simple change of basis. In the time domain, every sample of a signal represents its amplitude at that very instant.
\[ y\left …
Basis in a vector space
A vector space basis is the skeleton from which a vector space is built. It allows to decompose any signal into a linear combination of simple …
Inner product in vector space
The inner product is an operation that measures the similarity between vectors. In a general way, the inner product could be defined as an operation …
Properties of vector spaces
Vector spaces must meet the following rules: Addition to be commutative:
\( x + y = y + x \)
Addition to be distributive:
\( (x+y)+z = x + (y + z) \) …
Linear operator
A linear operator must meet the following conditions: \[ S\{\alpha x
\[n\]\} = \alpha S\{x
\[n\]\} \] \[ S\{x
\[n\]+ y
\[n\]\} = S\{x
\[n\]\} + …
Energy and power of a signal
\[ E_x = \sum_{n=-\infty}^{+\infty} \left| x\left
\[ n \\right\]\right|^2 \] \[ P_x = \lim_{N\to\infty} \frac{1}{2N + 1} \sum_{n = -N}^{N} \left| x …
Not all sinusoids are periodic in discrete time
For a signal to be periodic, it must fulfill the following condition:
\[ x
\[n\]= x
\[n+M\]\] where \(M \in \mathbb{Z}\).
If \(x
\[n\]= …
Onda estacionaria
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Resolución FFT
\[ \Delta f = \frac{F_s}{N_{samples}} \]
Si por ejemplo, \(F_s = 5GSa/s = 5\cdot10^9Sa/s \) y \(N_{samples} = 25000\), la resolución frecuencial es …