Problem Scenario
This problem will illustrate how powerful DFT-based processing can be in practice. Suppose you know that you have samples that come from a baseband real-valued periodic signal of an unknown period and that this signal is corrupted by additive white Gaussian noise. Suppose the period of the signal is unknown and you wish to measure it. Now, one option is to try the processing in the time-domain by measuring the time between two signal features - an example is shown in the next figure:
The problem is the following: if the signal is received at a low input SNR the time-domain characteristics may not even be visible, as illustrated for the above signal in the figure below:
Clearly, at this SNR it is impossible to measure the period in the time domain as described above. But if we compute the DFT and plot its magnitude it may be possible to detect (and then measure) the fundamental frequency - from which one can get the period. For this signal above, appropriate DFT processing would yield something like this:
So clearly DFT processing provides a powerful method here.
Problem Specifics (here is what you have to do!)
Assume that the periodic signal is an acoustic signal coming from an engine that is known to operate at rotational rate of between 5000 - 9000 rpm (revolutions per minute). Assume that the fundamental frequency of the signal corresponds to this rotational rate. Assume that the signal is sampled at the rate of 8000 samples/second. To make it easier to apply the results we derived, assume complex-valued additive white Gaussian noise. Assume that the power ratio between the signal fundamental's power and the noise power is -30 dB.
Note that since the signal is periodic you know from Fourier Series that it can be thought of as having a fundamental and harmonics. You can assume that the harmonics that lie above Fs/2 have been eliminated by an anti-aliasing filter.
Your job is to make sure the DFT processing will work by specifying:
• DFT processing length needed
• Window Choice
2. Consider the following scenario: You have a system intended to receive a sinusoidal RF signal whose frequency is unknown but is known to be one of the following 9 frequencies (all in GHz):
1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09
Note: spacing is 10 MHz
Assume that the signal is available as a LPE signal that has been sampled such that -Π corresponds to 1.0 GHz and +Π corresponds to 1.1 GHz.
Consider that you are going to process the LPE signal to detect the presence of the sinusoid in noise but that someone is trying to prevent you from doing so by "jamming" your receiver: they are going to transmit a signal in the 1.0 GHz to 1.1 GHz band that will mask the sinusoid to be detected.
Note that this is the same scenario as the first problem in Evaluation #1! So, you already know what the LPE signal looks like for this scenario.
Assume that your adversary has two options available for the jamming signal:
• A sum of 10 sinusoids of equal amplitude that are uniformly spread in frequency across the 1.0 GHz to 1.1 GHz band at the following frequencies (in GHz):
1.005 1.015 1.025 1.035 1.045 1.055 1.065 1.075 1.085 1.095
Note: spacing is 10 MHz
• A random signal that can be modeled as white and Gaussian Assume you know the following:
• The maximum received power of the desired sinusoid is 1 mW
• The receiver's internal noise power is negligible
• The maximum received power of the jamming signal is 1 W (Note: power of a sum of sinusoids is the sum of their powers) (Note: for a random signal, power = variance)
Your Task: Assuming that an auxiliary subsystem can detect which of the two jamming schemes is being used, define and test two DFT processing schemes (one for each of the two jamming signals) that can ensure detection of the desired sinusoid in the presence of the jammer
- obviously you need to strive to keep the number of samples used as small as possible. Your system will work directly on the LPE signal that you can assume is provided by some other processing that precedes your processing.