Tony Tang
EE222
Arya Akmal
Notes
Linear Transformation Cascaded Connection
Viewed as a linear transformation, the product AB represents the cascade (series)
connection of B followed by A, as shown below. This means that for a n-dimensional
input vector x
Tony Tang
EE222
Arya Akmal
Notes
The norm, or length, of a is dened by
and is strictly positive unless a = 0. Scaling a by a constant c results in |a| being scaled
by |c|.
Two vectors a and b are orthogonal (denoted by ab) if < a, b> = 0. This notion of
Tony Tang
EE222
Arya Akmal
Notes
Matrix Inversion:
The inverse of a square matrix A, denoted by A-1, is a square matrix with the property
A-1A = AA-1 = I,
where I is an identity matrix of the same size.
Useful properties of the inverse of a nonsingular
Tony Tang
EE222
Arya Akmal
Notes
Aliasing:
Two continuous-time sinusoids having dierent frequencies f and f (Hz) may, when
sampled at the same sampling rate fs, produce sample sequences having eectively the
same frequency. This phenomenon is known as ali
Tony Tang
EE222
Arya Akmal
Notes
Effective Frequency Range
The range of values
Ts [0,T/2] or equivalently fs [2f, ]
corresponds to [0,], which is the eective frequency range for real sinusoids.
If Ts > T/2, or equivalently, if fs < 2f, the sequence of s
Tony Tang
EE222
Arya Akmal
Notes
Key Complex Exponentials:
Sampling a continuous-time sinusoid at a rate of fs =1/Ts (samples/second) produces a
discrete-time sinusoid.
If the continuous-time sinusoid has angular frequency = 2f =2/T , the resulting
disc
Tony Tang
EE222
Arya Akmal
Notes
Complex Exponentials
Take a complex number z with modulus |z| = r and angle z = :
z = r(cos + j sin )
An alternative form for z is
z = rej
where we use the identity
ej = cos + j sin
This can be obtained from the Taylo
Tony Tang
EE222
Arya Akmal
Notes
Equations and Curves
The modulus |z| represents the length of the vector z. Similarly, the expression |z1-z2|
gives the distance between the points z1 and z2.
Using this concept, we see that the following two equations i
Tony Tang
EE222
Arya Akmal
Notes
Key Facts
Two complex numbers can be multiplied by expressing each number in the form z = x +
jy, then using distributivity and the rule j2 = -1 (i.e., j is treated as the square root of -1).
The product of two complex n
Tony Tang
EE222
Arya Akmal
Notes
Inner Product
For real-valued vectors, the dot product is also known as inner product and plays an
important role in the development of vector geometry. If a and b are real and mdimensional, their inner product is denoted
Tony Tang
EE222
Arya Akmal
Notes
Inner Products
The minimization can performed equivalently on the squared distance
where the last equality was obtained using the fact that VT V is a diagonal matrix with
each (j, j)th entry given by |v(j)|2 . It suces to