Question: Write a short note on different methods for digital signal synthesis.
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 modified 10 days ago by Abhishek Tiwari ♦♦ 50 written 12 months ago by
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Different methods for digital signal synthesis are:

Discrete Wavelet Transform (DWT):

Discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information (location in time).

Haar wavelet transform:

It may be considered to pair up input values, storing the difference and passing the sum. This process is repeated recursively, pairing up the sums to provide the next scale, which leads to $(2^n-1)$ differences and a final sum.

Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency domain representation.

The bilateral or two-sided Z-transform of a discrete-time signal x[n] is the formal power series X(z) defined as

$X(z) = Z{x[n]} = ∑_{n=-oo}^{oo} x[n]z^{-n}$

The inverse Z-transform is

$x[n] = Z^{-1} {X(z)} = \frac{1}{2nj} X(z)z^{n-1}dz$

where C is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC). In the case where the ROC is causal (see Example 2), this means the path C must encircle all of the poles of X(z).

Frequency domain:

Fourier transform decomposes a function of time (a signal) into the frequencies that make it up, similarly to how a musical chord can be expressed as the amplitude (or loudness) of its constituent notes. The Fourier transform of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is called the frequency domain representation of the original signal. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time

For a square image of size N×N, the two-dimensional DFT is given by:

Time domain:

The most common processing approach in the time or space domain is enhancement of the input signal through a method called filtering. Digital filtering generally consists of some linear transformation of a number of surrounding samples around the current sample of the input or output signal. There are various ways to characterize filters; for example:

(i) A "linear" filter is a linear transformation of input samples; other filters are "non-linear". Linear filters satisfy the superposition condition, i.e. if an input is a weighted linear combination of different signals, the output is a similarly weighted linear combination of the corresponding output signals.

(ii) A "causal" filter uses only previous samples of the input or output signals; while a "non-causal" filter uses future input samples. A non-causal filter can usually be changed into a causal filter by adding a delay to it.

(iii) A "time-invariant" filter has constant properties over time; other filters such as adaptive filters change in time.

(iv) A "stable" filter produces an output that converges to a constant value with time, or remains bounded within a finite interval. An "unstable" filter can produce an output that grows without bounds, with bounded or even zero input.