SC - Lezione 25 - Properties of the DFT, Discrete Convolution, Discrete Short-Time Fourier Transform

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SC - Lezione 25 §

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Checklist §

Checklist

Domande, Keyword e Vocabulary §

• Discrete convolution of two signals
• Parseval Theorem (2-norm)
• Length of the output signal
• Linear Discrete Convolution and Circular Discrete Convolution
• How to avoid the zero padding
• Basic facts about convolution
• Discrete Short-Time Fourier Transform
• Support of a function (and why we recall it)
• bins
• Spectogram

Appunti SC - Lezione 25 §

Nyquist Point §

Consider a vector of length with all components equal to 0 excepts the component, that is equal to That component is called the Nyquist Point.

The DFT of the unit vector at Nyquist point is a real vector of alternating and .

Parseval Theorem - 2-norm of input and output vector of the DFT §

the 2-norm of the input signal to the DFT: and the output signal , is related through the parseval theorem:

Convolution of two signals §

We have two signals (vectors), and , and we perform their convolution by taking the componentwise product of the Discrete Fourier Transform (DFT) of and .

Suppose has length and has length , where . Let and . Then, the convolution in the frequency domain is given by: where denotes the discrete convolution, and represents the componentwise multiplication.

The discrete convoluton, is defined as:

How can we set the length of ?

• Linear Discrete Convolution: The resulting vector has a length of , a zero padding is applied to the values of outside the range.
• Circular Discrete Convolution: In this case, is treated as periodic. Any shifts that move elements beyond the last position wrap around to the beginning, creating a circular or periodic structure. Formally the component is intented as: and the resulting vector is of length

When formally defined on infinite sequences rather than finite vectors, the discrete convolution becomes:

To avoid the zero padding: we reverse the vector , in this way is aligned to , is aligned to and so on. We then multiply the aligned components and sum them.

Basic facts about convolution §

Consider these two signals what is the output shape? It’s a trapezoid.

Discrete Short-Time Fourier Transform §

Recall the example from the previous lesson where we used the DFT to recover the 11 digits of a phone number. In that example, we attempted to achieve this manually by applying the DFT to a segment of the signal in the time domain.

There is a way to do this “automatically” by using the Discrete Short-Time Fourier Transform (STFT). The STFT divides the frequency range into bins.

Unlike the DFT, the STFT provides the time-varying frequency spectrum of the entire signal. It is defined as:

Where:

• is the input discrete signal,
• is the STFT of at time index and (angular) frequency ,
• is the discrete window function centered around the time . The window function is used to isolate a segment of the input signal .

Support of a Function: A function defined over is said to have support on an interval if it is non-zero only within that interval and zero elsewhere. This interval is the support of the function. Why we recall this fact? Because is usually a window function with finite support, so that the infinite sum in the formula becomes an infinite summation.

We skipped the detailed explanation of how this works.

Spectogram §

The Matlab function spectrogram computes the STFT of a signal. It returns a complex matrix whose columns contain an estimate of the short-term, time-localized frequency content of the input signal.

The magnitude squared of is the spectrogram time-frequency representation of the signal, that provides information about the distribution of the signal’s energy over time and frequency.

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