Question

Directions: The question consists of two statements, one labeled as ‘Statement (I)’ and the other labeled as ‘Statement (II)’. You are to examine these two statements carefully and select the answers to these items using the codes given below:

Statement (I): Non-stationary signals such as an image require time-frequency analysis.

Statement (II): The short-time Fourier transform (STFT) can map a one-dimensional function f (t) into the two-dimensional function, STFT (f).

1. Both Statement (I) and Statement (II) are individually true and Statement (II) is the correct explanation of Statement (I)
2. Both Statement (I) and Statement (II) are individually true but Statement (II) is NOT the correct explanation of Statement (I)
3. Statement (I) is true but Statement (II) is false
4. Statement (I) is false but Statement (II) is true

Answer 1

Correct Answer - Option 2 : Both Statement (I) and Statement (II) are individually true but Statement (II) is NOT the correct explanation of Statement (I)

• Stationary signals are signals whose spectral characteristics do not change with time. They are represented as a function of time or frequency.

• Non-stationary signals which involve both time and frequency, such as the image, requires time-frequency analysis.

• The time-frequency analysis involves mapping a signal which is a one-dimensional function of time, into an image which is a two-dimensional function of time and frequency. (Statement 1 is correct)

• The modification required in the Fourier transform is localizing the analysis so that it is not necessary to have the signal over (-∞, ∞) to perform the transform.

 

​STFT (Short-time Fourier transform) can map a one-dimensional function f(t) into the two-dimensional function STFT (f).

f(t) is assumed to be a stationary signal when viewed through a temporal window, t(t), which is a complex function.

Hence STFT (τ,f) is defined as:

\(STFT\left( {\tau ,f} \right) = \mathop \smallint \limits_{ - \infty }^\infty f\left( t \right){w^*}\left( {t - \tau } \right){e^{ - j2\pi ft}}dt\)

∴ Statement (2) is also correct but is not the correct explanation of Statement (1).