1. tansform
The mathematical term: transform, is extensively used in Digital Signal
Processing, such as: Fourier transform, Laplace transform, Z transform,
Hilbert transform, Discrete Cosine transform, etc.
Just what is a transform?
To answer this question, remember what a function is. A function is an
algorithm or procedure that changes one value into another value. For example,
y=2x+1 is a function. You pick some value for x, plug it into the equation,
and out pops a value for y. Functions can also change several
values into a single value, such as: y=2a + 3b + c , where a, b and c are
changed into y.
Transforms are a direct extension of this, allowing both the input and output
to have multiple values. Suppose you have a signal composed of 100 samples.
If you devise some equation, algorithm, or procedure for changing these 100
samples into another 100 samples, you have yourself a transform. If you think
it is useful enough, you have the perfect right to attach your last name to it
and expound its merits to your colleagues. (This works best if you are an
eminent 18th century French mathematician). Transforms are not limited to any
specific type or number of data. For example, you might have 100 samples of
discrete data for the input and 200 samples of discrete data for the output.
Likewise, you might have a continuous signal for the input and a continuous
signal for the output. Mixed signals are also allowed, discrete in and
continuous out, and vice versa. In short, a transform is any fixed procedure
that changes one chunk of data into another chunk of data.
2. impulse decomposition and Fourier decomposition
the fundamental concept of DSP: the input signal is decomposed into simple
additive components, each of these components is passed through a linear
system, and the resulting output components are synthesized (added). The
signal resulting from this divide-and-conquer procedure is identical to that
obtained by directly passing the original signal through the system. While
many different decompositions are possible, two form the backbone of signal
processing: impulse decomposition and Fourier decomposition. When impulse
decomposition is used, the procedure can be described by a mathematical
operation called convolution.
3. Fourier transform
any periodical signal can be represented as a linear combination of a set of
sinusoidal signals of different frequencies, amplitudes, and phase angles
3.1
Acos(x) + Bsin(x) = M*cos( x + θ)
M=sqrt(A^2 + B^2)
θ=arctan(B/A)
4. complex
Rectangular complex ==> polar complex
M=sqrt(a^2 + b^2)
θ=arctan(b/a) #if a>0
θ=arctan(b/a) + p #if a<0
polar complex ==> Rectangular complex
a=Mcosθ
b=Msinθ
e^jx = cos(x) + jsin(x) #Euler’s relation
a + jb <==> M ( cosθ + jsinθ )
a + jb <==> Me^(jθ) #plug in Euler relation
rectang vs polar
5. how to use complex to solve real problem
The idea to remember is that some physical problems can be converted into a
complex form by simply adding a j to one of the components. Converting back to
the physical problem is nothing more than dropping the j. This is the essence
of the substitution method.
6. Complex Representation of Sinusoids
能使用复数表示sine波的前提条件是：1.所有的sine波频率一样 2.系统是线性系统
Acos(ωt) + Bsin(ωt) <==> a + jb #where A=a, B=-b
(conventional representation) (complex number)
Mcos(ωt + N) <==> Me(jθ) #where M=M, θ=-N
(conventional representation) (complex number)