Discrete Fourier Transform (DFT) Matching

[roam]

Homework Statement

Match each discrete-time signal with its DFT:

Homework Equations

The Attempt at a Solution

I am mainly confused about Signal 7 and Signal 8.

Signal 1 is the discrete equivalent to a constant function, therefore its DFT is an impulse (Dirac $\delta$), so it corresponds to DFT 3.

DFT of an impulse is a constant. Therefore Signal 6 corresponds to DFT 5.

Signal 2 is a sampled version of a full period of a cosine. So we expect $X(1)$ and $X(-1)$ to be nonzero ($X(-1)$ really is $X(N-1)=X(25)$). Therefore Signal 2 corresponds to DFT8.

By similar arguments, Signal 4 has exactly 2 cycles of a cosine and corresponds to DFT2. And Signal 3 has one and a half periods of a cosine, as we do not have complete periods we should expect spectral leakage, so signal 3 corresponds to DFT4 (the main peaks are around 1 & 2 plus negative frequencies). Likewise, Signal 5 corresponds to DFT7.

Here is a summary of the results so far:

Only Signals 7 & 8 and DFT 6 & 1 are left:

What do signals 7 and 8 represent? Is Signal 7 an undersampled cosine? How do we go about matching them with their corresponding DFTs?

Any explanation would be greatly appreciated.

 

[DrClaude]

Could you please clarify what the scale of the DFT represents? Is it the the same order as the output of an FFT (from 0 to highest positive frequency, followed by lowest negative frequency back to 0)?

What do signals 7 and 8 represent? Is Signal 7 an undersampled cosine? How do we go about matching them with their corresponding DFTs?

Both can be see as undersampled cosines, with Signal 7 being a cosine with a varying amplitude envelope.

 

[roam]

Could you please clarify what the scale of the DFT represents? Is it the the same order as the output of an FFT (from 0 to highest positive frequency, followed by lowest negative frequency back to 0)?

It is similar to the spectrum of an FFT which is not fftshifted. The zero-frequency component (DC) is the first element ($r=0$). Then it is the positive frequencies, but I think it is lowest to largest, followed by negative frequencies.

Both can be see as undersampled cosines, with Signal 7 being a cosine with a varying amplitude envelope.

Yes, this is right. Any ideas how to identify the DFT for each signal?

 

[DrClaude]

Any ideas how to identify the DFT for each signal?

What case would correspond to a single frequency?

 

[roam]

What case would correspond to a single frequency?

Is it Signal 8?

So, the spectrum of Signal 8 is DFT6? What would be a good explanation? Signal 7 looks like at least two cosine waves being heterodyned (i.e. a cosine wave contained in a lower frequency cosine envelope).

 

[DrClaude]

So, the spectrum of Signal 8 is DFT6? What would be a good explanation? Signal 7 looks like at least two cosine waves being heterodyned (i.e. a cosine wave contained in a lower frequency cosine envelope).

That's basically it. Signal 8 corresponds to a pure cosine sampled twice per oscillation period, and therefore has a single frequency. The effect of changing the envelop (modulating the amplitude) is a spreading out of the frequency, as given by DFT1 (you can see it as an "uncertainty" in the frequency due to the limited time, or pulse nature, of the signal, or due to the fact that the signal is similar to a beat coming from overlapping signals of similar frequencies).