4.12  Exercises

4.1. Using the DTFT $W(e^{j\mathrm{\Omega }})$ of the $N$-samples rectangular window given in Eq. (4.7), 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.
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?
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?
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?
4.5. Assume $X[k]$ is the $N$-points 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-points FFT.
4.6. Assume $X=[8,8,8,8]$ is the $4$-points 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 $P$.
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 $Ŝ[1]$?
4.8. The periodogram of a sinusoid immersed in AWGN was calculated with an $8$-points 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.
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.
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)