The Scientist and Engineer's Guide to Digital Signal Processing一書的閱讀總結

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

A*cos(x) + B*sin(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

能使用複數表示sine波的前提條件是：1.所有的sine波頻率一樣 2.系統是線性系統

       A*cos(ωt) + B*sin(ωt) <==> a + jb #where A=a, B=-b

(conventional representation)    (complex number)

            M*cos(ωt + N) <==> M*e(jθ) #where M=M, θ=-N

(conventional representation) (complex number)