4.14  Exercises

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4.1.

Using the DTFT $W(e^{j\mathrm{\Omega }})$ of the $N$-samples rectangular window given in Eq. (4.9), prove that the DTFTs of the Hann and Hamming windows, given in Eq. (4.4) and Eq. (4.3), are $0.5W(e^{j\mathrm{\Omega }})-0.25W(e^{j(\mathrm{\Omega }+2\pi ∕(N-1))})-0.25W(e^{j(\mathrm{\Omega }-2\pi ∕(N-1))})$ and $0.54W(e^{j\mathrm{\Omega }})-0.23W(e^{j(\mathrm{\Omega }+2\pi ∕(N-1))})-0.23W(e^{j(\mathrm{\Omega }-2\pi ∕(N-1))})$, respectively.

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4.2.

A cosine $x[n]=A\text{cos}<!--\; FUNCTION\; APPLICATION\; -->({\mathrm{\Omega }}_{1}n)$ is multiplied by a rectangular window of $N=8$ samples, from $n=0,\dots ,7$ and the windowed signal $x_w[n]$ has its spectrum estimated by an FFT with $N$ points. The FFT resolution is $\mathrm{\Delta }\mathrm{\Omega }=(2\pi )∕N$ rad. Using the DTFT of $x_w[n]$, what are the FFT values for $X[k]{|}_{k=3}$ assuming: a) ${\mathrm{\Omega }}_1=3\mathrm{\Delta }\mathrm{\Omega }$ is centered in the fourth bin and b) ${\mathrm{\Omega }}_1=2.5\mathrm{\Delta }\mathrm{\Omega }$ is half-way the third and fourth bins?

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4.3.

Given the signal $x(t)=12\mathrm{\text{sinc}}(6t)$ in volts, what are its: a) total signal energy and b) energy within the frequency band $[-1,12]$ Hz?

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4.4.

What is the MS spectrum ${Ŝ}_{\mathrm{\text{ms}}}[k]$ of $x[n]=3\text{cos}<!--\; FUNCTION\; APPLICATION\; -->((\pi ∕4)n)$ volts? Using the values of ${Ŝ}_{\mathrm{\text{ms}}}[k]$, how can one obtain the average power of $x[n]$ in watts?

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4.5.

Assume $X[k]$ is the $N$-point FFT of a periodic signal $x[n]$ of period $N_0=8$ samples. When $N=8$, $X[k]$ is $[0,3+4j,0,6,0,6,0,3-4j]$. Inform: a) the estimated MS spectrum ${Ŝ}_{\mathrm{\text{ms}}}[k]$ of $x[n]$, b) the signal power $P$ and c) the new ${Ŝ}_{\mathrm{\text{ms}}}[k]$ in case $N=16$ samples of $x[n]$ were used, together with a 16-point FFT.

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4.6.

Assume $X=[8,8,8,8]$ is the $4$-point FFT of a discrete-time signal $x[n]$. What are the values of its: a) DTFS, b) MS spectrum ${Ŝ}_{\mathrm{\text{ms}}}[k]$, c) the periodogram $Ŝ[k]$ (using Matlab/Octave convention of $\mathit{BW}=2\pi$), d) the estimated PSD $Ŝ(e^{j\mathrm{\Omega }})$ and e) the signal power $P$.

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4.7.

a) What is the PSD $S(e^{j\mathrm{\Omega }})$ of a complex exponential $x[n]=e^{j{\mathrm{\Omega }}_{1}n}$, with ${\mathrm{\Omega }}_1=\pi ∕4$ rad after multiplication by a rectangular window of $N=10$ samples? b) When the periodogram $Ŝ[k]$ of this signal is estimated with a FFT of $N=10$ points, what are the values of $Ŝ[0]$ and $Ŝ[1]$?

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4.8.

The periodogram of a sinusoid immersed in AWGN was calculated with an $8$-point FFT as $[2,2,2,6,2,6,2,2]$ in watts/Hz, assuming $F_s=500$ Hz. The noise and the sinusoid are uncorrelated, such that, at the sinusoid bins, the power is the sum of the sinusoid power and the noise power at that bin. Inform: a) the sinusoid average power, b) the noise average power, c) the SNR in dB.

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4.9.

The bilateral PSD of a continuous-time white noise signal $\nu (t)$ is $N_0∕2=8$ W/Hz. This signal was digitized using an ideal lowpass filter and $F_s=20$ kHz, creating a discrete-time signal $\nu [n]$. Inform: a) the average power of $\nu (t)$, b) the average power of $\nu [n]$ and c) the power corresponding to a single periodogram bin of a windowed $\nu [n]$ estimated with an FFT of length $N=256$ when $F_s=20$ kHz and $F_s=2\pi$ are informed.

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4.10.

Assuming $F_s=100$ Hz, the result of Welch’s method with 8-length FFTs for a real signal composed by a sum of two sinusoids was S = [0,100,0,20,0] in dBW/Hz. Inform: a) the sinusoid frequencies, b) the sinusoid powers in watts and c) their power ratio in dB. Hint: one can use Matlab/Octave to investigate this setup with the commands:

1N=8; n=0:N-1; x1=4*cos(pi/4*n); x2=10*cos(3*pi/4*n); x=x1+x2; 
2Fs=100; [S,f]=pwelch(x,rectwin(N),0,N,Fs), S*f(2), SdB=10*log10(S)

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4.11.

A signal $x[n]$ has its PSD estimated via AR modeling with the result: A=[1.0, -1.8151, 0.9025] and Perror=4 (for example, with the Matlab command [A,Perror]=lpc(x,2)). What is the frequency of the peak of this PSD and its amplitude?

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4.12.

An autoregressive model $H(z)$ of order one was obtained with the command [A,Perror]=lpc(x,1)), where A=[1, -0.75] and Perror=0.04 watts. a) What is the expression for $H(z)=g∕A(z)$ assuming that $g$ incorporates the information from Perror? b) What is the expression for the PSD $Ŝ(f)$ corresponding to this model assuming $F_s=100$ Hz (the expression must depend only on $f$)? c) What is the value of $Ŝ(f)$ at frequency $f=2$ Hz?

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4.13.

The unilateral PSD of a continuous-time white noise signal $\nu (t)$ is $N_0=4$ W/Hz. The signal $\nu (t)$ is the input to a lowpass filter with real-valued coefficients, gain $g=3$ and zero phase within the passband from 0 to 200 Hz (this filter is not realizable). The filter output $x(t)$ is converted to a discrete-time signal $x[n]$ using an A/DT process with sampling frequency $F_s=1$ kHz. The average power values of $x(t)$ and $x[n]$ are the same and denoted as $P$. Inform: a) the value $P$ in watts, b) the detailed graphs of the PSDs $S_{\nu }(f)$, $S_x(f)$, and $S_x(e^{j\mathrm{\Omega }})$ corresponding to the signals $\nu (t)$, $x(t)$ and $x[n]$, respectively. Note that $x[n]$ is not a discrete-time white noise, because its PSD does not have a constant value over the range $[-F_s∕2,F_s∕2[$.

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4.14.

The periodogram of a signal $x[n]$ with $N=100$ samples is estimated via Welch’s method using a rectangular window $w[n]$ with $M=25$ samples. The sampling frequency is $F_s=1∕T_s=100$ Hz. The window shift is $M$ samples, such that there is no overlapping among windows, and the samples of $x[n]$ are organized into four blocks of $M$ samples each. After zero-padding, each block of samples is converted to frequency domain using an FFT of $N_f=128$ points. a) What is the frequency spacing ${\mathrm{\Delta }}_f$ in Hertz between neighboring periodogram bins? b) What is the frequency resolution ${\mathrm{\Delta }}_m$ in Hertz imposed by the DTFT $W(e^{j\mathrm{\Omega }})$ of the windows $w[n]$? Assume that ${\mathrm{\Delta }}_m$ is the range between the two zeros of $W(e^{j{T}_{s}2\mathit{\pi f}})$ that define its main lobe, where $W(e^{j{T}_{s}2\mathit{\pi f}})$ is the DTFT converted to continuous-time using $\mathrm{\Omega }=T_s\omega =T_{s}2\mathit{\pi f}$. c) Is the overall frequency resolution limited by ${\mathrm{\Delta }}_f$ or ${\mathrm{\Delta }}_m$? d) Why is it that someone cannot consistently improve the overall resolution in spectral analysis by simply using zero-padding and improving ${\mathrm{\Delta }}_f$ by using a larger number $N_f$ of FFT points?