Additional Examples of Chapter 4:
Digital Signal Processing of Continuous-Time Signals
Example E4.1: A continuous-time signal x a (t) is composed of a linear combination of
sinusoidal signals of frequencies 250 Hz, 450 Hz, 1.0 kHz, 2.75 kHz, and 4.05 kHz.
EE608A: Least Squares
Stevens Institute of Technology
September 8, 2016
For any vector x Rn and vector y Rm , the matrix differentiation calculus is defined
Digital Processing of Continuous-Time Signals
Digital processing of a continuous-time signal involves
the following basic steps:
Conversion of the continuous-time signal into a
Discrete-Time Fourier Transform
Definition - The discrete-time Fourier transform
(DTFT) X (e j ) of a sequence x[n] is given by
x[n]e j n
The DTFT is often called the Fourier spectrum
In general, X (e j ) is a complex function of the
For up-sampler, see Example 2.17 (p. p75) of text.
Sampling rate is Fs = 2400 Hz for filter I(z)
NI=20 in above calculation. If using Matlab rezord, the filter orders are NF=391 and
The transfer function of a lowpass half-band filter can be expressed as
Since h[2n]=0 for non-zero n, and z=-1 is a zero of H(z), we have
Equivalently, we have
where in the second equality, we used again the fact that h[2n]=0 for non-zero n.
Note: The equation immediately following the figure has a sign error. The 2nd row of 1st matrix
should be -2cos(theta) 2r sin(theta); after inverse, the minus sign in 2nd row should be removed,
i.e., 1/(2r sin(theta) without the minus sign.
Using Eq. (2.9), we get:
To show this, we start with the definitions from Eq. (2.9) and square them:
The middle inequality is a generalization of the triangle inequality. We can take square
roots of bo