I want these problems of DSP to be solved for a meeting.
1.2 Tests to be carried out
1.2.1 Implementation of an adaptive filter
Implement an adaptive filter as shown in Figure 1.1 using the update equation 1.18 as a Matlab function. Pass the following arguments to the function: input signal x(n), desired signal d(n), filter length L and the step size parameter (ALPHArel) as relative size to the maximum stable step size according to calibration 1.33 The function shall calculate and return the following sizes: Error signal e(n), output signal y(n), time sequence of the set filter coefficients fn ( Attention: Represents a matrix)
1.2.2 Checking the Implementation: Notch-Filter
to check the self written function use the sum of a white noise and a sine tone at 1KHz as input signal x(n). As desired signal d(n) use the identical noise as in x(n), but without the sine tone. In total, the signal should be about 1s long (about 10,000 samples, depending on the selected sampling rate). Adapt the adaptive filter with this signal for different relstive step sizes (ALPHArel) between 0.1 and 1 and a filter length of L= 100 and document the instantaneous power of the noise e(n)^2 as a function of time to check the convergence. Further document the transfer function of the adaptive filter (to be calculated from the filter coefficient f(n) after convergence of the adaptive filter). Since in this circuit the input signal contains a sine tone that the desired signal does not contain, the adaptive filter should attenuate this frequency to equalize both signals. At all other frequencies, however, the two signals are identical, i.e. the filter should leave all other frequencies unchanged in order to equalize both signals. The result is a filter that attenuates exactly one frequency and leaves the rest unchanged, a so-called notch filter.
1.2.3 System detection
Connect the daptive filter as shown in Figure 1.2. Use an IIR low pass filter as the system to be detected (e.g. the design from the first week of the internship) and white noise as the input signal x(n). Let the adaptive filter adapt with a relative increment (ALPHArel) of about 0.8 and document the filter coefficients f(n) after convergence of the adaptive filter. Compare the transfer function of the system to be detected (the poles of the system function of the IIR filter are known) and of the adaptive filter by magnitude and phase.
1.2.4 Noise suppression
Use the same structure as for system detection and, in addition to the desired signal d(n), a speech signal u(n). After adaptation of the adaptive filter, the error signal e(n) should only contain the speech signal. Listen to the error signal and describe the hearing impression. If it meets the expectation