(i) Dual tone Multi-frequency Signal Detection.

Dtmf:

• Dual Tone Multi-Frequency, or DTMF is a method for instructing a telephone switching system of the telephone number to be dialed, or to issue commands to switching systems or related telephony equipment.

• The DTMF dialing system traces its roots to a technique AT&T developed in the 1950s called MF (Multi-Frequency) which was deployed within the AT&T telephone network to direct calls between switching facilities using in-band signaling.

• The DTMF system uses eight different frequency signals transmitted in pairs to represent sixteen different numbers, symbols and letters. This table shows how the frequencies are organized:

• The frequencies used were chosen to prevent any harmonics from being incorrectly detected by the receiver as some other DTMF frequency.

• The transmitter of a DTMF signal simultaneously sends one frequency from the high-group and one freqency from the low-group. This pair of signals represents the digit or symbol shown at the intersection of row and column in the table. For example, sending 1209Hz and 770Hz indicates that the "4" digit is being sent.

• At the transmitter, the maximum signal strength of a pair of tones must not exceed +1 dBm, and the minimum strength is -10.5 dBm for the low-group frequencies and -8.5 dBm for the high-group frequencies.

Labeling of DTMF numeric digits

• The DTMF telephone keypad is laid out in a 4×4 matrix of push buttons in which each row represents the low frequency component and each column represents the high frequency component of the DTMF signal. Pressing a key sends a combination of the row and column frequencies.

• For example, the key 1 produces a superimposition of tones of 697 and 1209 hertz (Hz). Initial pushbutton designs employed levers, so that each button activated two contacts. The tones are decoded by the switching center to determine the keys pressed by the user..

• DTMF was originally decoded by tuned filter banks. By the end of the 20th century, digital signal processing became the predominant technology for decoding. DTMF decoding algorithms often use the Goertzel algorithm to detect tones.

ii) Different methods for digital signal Synthesis

Methods:

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 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

$X(z)=Z\{x[n]\}=\sum\limits_{n=-\infty}^{\infty}x[n]z^{-n}$

The inverse Z-transform is

$x[n]=Z^{-1}\{X(z)\}=\dfrac 1{2\pi j} \oint_c X(z)z^{n-1} dz$

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:

The 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:

$F(k,l)=\sum\limits_{i=0}^{N-1}\sum\limits_{j=0}^{N-1} f(i,j)e^{-2\pi(\frac {ki}N+\frac {lj}N)}$

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:

• 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.

• 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.

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

• 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.