05-09-2012, 05:17 PM
Discrete time Fourier Series Analysis
4_DTFT.ppt (Size: 2.19 MB / Downloads: 38)
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 linear combination of scaled and delayed unit samples.
We can also represent any arbitrary discrete signal as a linear combination of basis signals introduced in Chapter 2.
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 the complex exponential signal set and is called the discrete-time Fourier Transform.
Matlab Implementation
If x(n) is of infinite duration, then Matlab can not be used directly to compute X from x(n).
We can use it to evaluate the expression X over [0,pi] frequencies and then plot its magnitude and angle (or real and imaginary parts).
For a finite duration, the DTFT can be implemented as a matrix-vector multiplication operation.
w: continuous--discrete
The properties of the DTFT
1. Linearity:
The DTFT is a linear transformation.
2. Time shifting:
A shift in the time domain corresponds to the phase shifting.
3. Frequency shifting:
Multiplication by a complex exponential corresponds to a shift in the frequency domain.
4. Conjugation:
Conjugation in the time domain corresponds to the folding and conjugation in the frequency domain.
4_DTFT.ppt (Size: 2.19 MB / Downloads: 38)
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 linear combination of scaled and delayed unit samples.
We can also represent any arbitrary discrete signal as a linear combination of basis signals introduced in Chapter 2.
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 the complex exponential signal set and is called the discrete-time Fourier Transform.
Matlab Implementation
If x(n) is of infinite duration, then Matlab can not be used directly to compute X from x(n).
We can use it to evaluate the expression X over [0,pi] frequencies and then plot its magnitude and angle (or real and imaginary parts).
For a finite duration, the DTFT can be implemented as a matrix-vector multiplication operation.
w: continuous--discrete
The properties of the DTFT
1. Linearity:
The DTFT is a linear transformation.
2. Time shifting:
A shift in the time domain corresponds to the phase shifting.
3. Frequency shifting:
Multiplication by a complex exponential corresponds to a shift in the frequency domain.
4. Conjugation:
Conjugation in the time domain corresponds to the folding and conjugation in the frequency domain.