文档介绍：Chapter 3. The Discrete-Time Fourier Analysis
Gao Xinbo
School of ., Xidian Univ.
Xbgao@
./teach/matlabdsp/
Introduction
A linear and time-invariant system can be represented using its response to the unit sample sequence.
h(n) is called as the unit impulse response
y(n)=x(n)*h(n): system response
The convolution representation is based on the fact that any signal can be represented by a bination of scaled and delayed unit samples.
We can also represent any arbitrary discrete signal as a bination of basis signals introduced in Chapter 2.
Introduction (con’t)
Each basis signal set provides a new signal representation.
Each representation has some advantages and disadvantages depending upon the type of system under consideration.
When the system is linear and time-invariant, only one representation stands out as the most useful. It is based on plex exponential signal set and is called the discrete-time Fourier Transform.
The discrete-time Fourier transform (DTFT)
DTFT:
IDTFT:
Existence Condition: x(n) is absolutely summable.
DTFT
F[.] transforms a discrete signal x(n) into plex-valued continuous function X of real variable w, called a digital frequency, which is measured in radians.
Time domain -- Frequency domain
Discrete -- Continuous
Real valued -- Complex-valued
Summation -- integral
The range of w:
The integral range of w:
Examples
Determine the discrete-time Fourier transform of
Result visualization with Matlab
Complex function: magnitude and angle;
real/imaginary part with respect to w
The range of w:
showing interval: [0,pi]
Two important properties
Periodicity:
The DTFT is periodic in w with period 2pi
Implication: we need only one period for analysis and not the whole domain
Symmetry:
For real-valued x(n), X is conjugate symmetric.
Implication: to plot X, we now need to consider only a half of X: [0,pi]
Symmetry
Matlab Implementation
If x(n) is of infinite duration, then Matlab can not be used direc