  This paper focuses on recognizing dialed numbers in a touch-tone telephone
system based on the Dual Tone MultiFrequency (DTMF) signaling technique with
analysis of stochastic aspects during the noise and random duration of
characters. Each dialed digit's acoustic profile is derived from a composite of
two carrier frequencies, distinctly assigned to represent that digit. The
identification of each digit is achieved by pinpointing the frequency pair with
the highest energy or amplitude in its spectral output, utilizing the
Discrete-Time Fourier Transform (DTFT). This analysis includes simulations that
illustrate the effects of introducing stochastic variations during the "Mark"
and "Space" intervals of the decoding process, offering insights into the
technique's efficacy and the impact of random temporal fluctuations. This will
reduce the accuracy of decoder decoding and lower the SNR of the system.
