For the purpose the Z-transform, which is the best known from the family of discrete transformations is taken. The order of the difference equation (0.3) is the positive integer k which is the largest difference in the index of nonzero values of y. The general solution of an inhomogeneous linear difference equation with constant coefficients (0.3) is the sum of the complementary function and any particular solution.
CHAPTER-2 Z-TRANSFORM
The Direct z-transform
The problem of finding the ROC for X(z) is equivalent to determining the range of values of r for which the sequence ( ) is absolutely summable. The ROC for the second sum therefore consists of all points outside a circle with radius>.
Importance Properties of the ROC for the z-transform (i) The ROC does not contain any poles
- Definition and properties
Since the convergence of X(z) requires that both sums be finite, it follows that the ROC of both sums are finite. The one-sided z-transform differs from the two-sided transformation in the lower bound of the summation, which is always zero regardless of whether the signal x(n) is zero or not for n=0 (i.e. causal).
The Inverse z-transform
- Power series method
- Residue method
In this form, the inverse -transform, ( ), can be obtained using one of several methods, including the following three:. 2) Partial fraction expansion method, (3) Residual method. In this method, the numerator and denominator of ( ) are first expressed in either descending powers of ascending powers of and the quotient is then obtained by long division. If the order of the numerator is less than that of the denominator in equation (2.16), i.e.
The coefficient associated with the pole can be obtained by multiplying both sides of equation 4.15 by ( − )⁄ and. If For example, if X(z) contains a pole of mth order at =, the partial fraction expansion must contain terms of the form.
For rational polynomials, the contour integral in equation (2.19) is evaluated using a fundamental result in complex variable theory known as Cauchy's residue theorem. In the last section it was stated that the partial fraction coefficients, the , are also referred to as residues of X(z) and a way of obtaining their values was given.
PROPERTIES OF Z-TRANSFORM (i) Linearity
If we take a parameter, then we can differentiate both sides with respect to a, so we. So Z na n1 can be obtained using the multiplication effect of n. Continuing the process m times and after simplification we will get the result. If both conditions are not met simultaneously, then we will not get the required transformation.
Both conditions can be satisfied only if a b , so z will lie within an annular region.
CHAPTER-3
DISCRETE FOURIER TRANSFORM AND FAST FOURIER TEANSFORM
The Discrete-Time Fourier Transform
- Connection between the DTFT and the Fourier Transform
- The DFS and the DFT
The Discrete-Time Fourier Transform describes the spectrum of discrete-time signals and formalizes the concept that discrete-time signals have periodic spectra. Ideal sampling of an analog signal x t leads to the ideal sampled signal xI t whose spectrum Xp f is periodic. They allow us to obtain the periodic spectrum Xp f of an ideal sampled signal from its samplesx nt s , and to recover the samples x nt s from the spectrum.
We emphasize that these relations are exact duals of the Fourier series relations for the periodic signal xp t and its discrete spectrum X k (Fourier series coefficients). These relations can be corrected for discrete-time signals by using the digital frequency F f /S and replacing x nt s with the discrete sequence x n to obtain. Sampling and duality provide the basis for the connection between all frequency domain transformations, and the concept is worth repeating.
The sample distance in one domain is the reciprocal of the period in the other domain. This leads to the development of the discrete Fourier transform (DFT) and the discrete Fourier series (DFS), giving us a practical way to arrive at the sampled spectrum of sampled signals using digital computers.
DFT and IDFT
- The Inverse DFS
Operation in the Time Domain Result in the Frequency Domain Transform Aperiodically continuous x t Aperiodically continuously X f FT. As a result, the DFT and its inverse are also periodic with period N, and it is sufficient to calculate the results for only one period (0 to N-1). Here is an example to calculate the DFT of a series and get it back using IDFT.
Since only x(0) and x(1) are non-zero, the top index in the DFT summation will be n=1 and the DFT reduces to. The quantity XDFS( )k defines the discrete Fourier series (DFS) as an approximation to the Fourier series coefficients of a periodic signal and is equal to N times the DFT. The sampling interval ts does not enter into the calculation of the DFS or its inverse.
Matrix representation of DFT
To recover x n( ) from one period XDFS( )k, we use a Fourier series reconstruction whose summation index covers one period (from k=0 to k=N-1) to obtain. We note that the computation of each DFT point can be achieved with N complex multiplications and ( − 1) complex additions. Therefore, the values of DFT -points can be calculated in the total sum of….complex multiplications and ( − 1)complex additions.
If we assume that the inverse of exists, (3.3.4) can be reversed by pre-multiplying both sides by. Naturally, the existence of the converse of previously was established based on our derivation of the IDFT. The IDFT of can be determined by conjugating the elements to obtain ∗ and applying the formula (3.3.5).
Relationship of the DFT to other transforms .1 Relationship to the z-transform
- Relationship to the Fourier series coefficients of a continuous-time signal
If the sequence ( ) has a finite duration of length or less, the sequence can be retrieved from its -point DFT. When evaluated on the unit circle, (3.4.3) yields the Fourier transform of the finite-duration sequence in terms of its DFT, in the form This expression for the Fourier transform is a polynomial (Lagrange) interpolation formula for ( ) expressed in terms of the values { ( )} of the polynomial at a set of equally spaced discrete frequencies.
Clearly, the above equation is in the form of an IDFT formula, where.
Efficient Computation of the DFT: FFT Algorithms
- Direct Computation of the DFT
- Divide-and-conquer approach to computation of the DFT
- Radix-2 FFT Algorithms
Direct DFT calculation is essentially inefficient mainly because it does not take advantage of the symmetry and periodicity properties of the phase factor. Then the DFT can be expressed as the double sum of the elements of the orthogonal array multiplied by the corresponding phase factors. At first glance, the computational procedure described above may seem more complicated than a direct DFT calculation.
So, instead of having to perform 10 complex multiplications via direct calculation of the DFT, this. The final step is to calculate the five-point DFTs for each of the three columns. This calculation gives the desired values of DFT in the form. In the previous section, we described two algorithms for efficient computation of the DFT based on the divide-and-conquer approach.
In such a case, the DFTs are of magnitude such that the computation of the −point DFT has a regular pattern. The first step results in a reduction in the number of multiplications from To ⁄ +2 ⁄2 , which is about a factor of 2 too large.
Some Practical Guidelines
The decimation of the data sequence can be repeated over and over until the resulting sequences are reduced to one-point sequences. The defining relation for the DFT (or DFS) requires that samples of ( ) be chosen over the range through periodic extension if necessary). If a sampling instant corresponds to a jump discontinuity, the sample value must be chosen as the midpoint of the discontinuity.
The calculation of the DFT (or DFS) is independent of the sampling frequency S or sampling interval ts 1/S. To compare the DFT results with conventional two-sided spectra, it should be remembered that by periodicity, a negative frequency − (at the index − ) in the two-sided spectrum corresponds to the frequency − (at the index − ) in the (one-sided ) DFT spectrum. The highest frequency in the DFT spectrum corresponds to the fold index = 0.5 and equal to f 0.5S Hz for the sampled analog signals.
The DFT (or DFS) can also be plotted as a two-sided spectrum to reveal the conjugate symmetry about the origin by creating its periodic expansion. This is equivalent to creating a rearranged spectrum by moving the DFT patterns at indices past the folding index k=0.5N to the left.
CHAPTER-4 WAVELET TRANSFORM
WINDOW FUNCTION
A figure of merit for the time-frequency window is its frequency-time-width product ∆ ∆, which is bounded below by the uncertainty principle and is given by.
DISCRETE SHORT-TIME FOURIER TRANSFORM
CONTINUOUS WAVELET TRANSFORM
- Inverse Wavelet transform
If we choose the parameters of the window function ( ) [ in the case ofg ( )] so that. In approximately equal to AB, the STFT as calculated using (4.2.1) will be able to better resolve the low-frequency portion of the signal, while there will be poor resolution of the high-frequency portion . On the other hand, if ∆ is approximately equal to CD, the low frequency will not be properly resolved.
Note that if ∆ is very small, ∆ will be relatively large, so the low frequency part will be blurred. In other words, we need to have a window function whose radius increases with time (decrease in frequency) while resolving the low-frequency content, and folds in time (increase in frequency) while resolving the high-frequency content of the signal. Since the purpose of the inverse transform is to reconstruct the original signal/function from its transformed form, in the case of the integral wavelet transform it involves a two-dimensional integration over a scale parameter a and a translation parameter b.
Integration with respect to a sums up all the combinations of the wave components at location b, while the integral with respect to b includes all locations along the b axis. Since calculation of the inverse wavelet transform is quite cumbersome and the inverse wavelet transform is only used for synthesizing the original signal, it is not used as f.
Books Consulted
