Question 7 of 10: A Nonlinear/Filter Cascade — Linearity, Distortionless Transmission, and Black-Box Testing

Question 7: A Nonlinear/Filter Cascade — Linearity, Distortionless Transmission, and Black-Box Testing (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. $x(t)=2\cos2\pi f_0t$; nonlinear block: output $=$ input $+$ input$^2$; lowpass filter $H(f)$ with the trapezoidal magnitude and piecewise-linear phase described above.

Find. (a) $X(f),Y(f)$; (b) $Z(f),z(t)$; (c) linearity/distortionlessness of the cascade; (d) black-box test design.

Approach. Expand the square with $\cos^2\theta=\tfrac12(1+\cos2\theta)$ to get $Y(f)$ as three impulse pairs (DC, $f_0$, $2f_0$), then evaluate $H(f)$ (magnitude AND phase) at exactly those three frequencies to find which survive into $Z(f)$.

1. Part (a) — X(f), Y(f). $X(f)=\delta(f-f_0)+\delta(f+f_0)$ (weight 1 each, since $x(t)=2\cos2\pi f_0t$). With $\theta=2\pi f_0t$, $$y(t)=x(t)+x(t)^2=2\cos\theta+4\cos^2\theta=2\cos\theta+2(1+\cos2\theta)=2+2\cos\theta+2\cos2\theta,$$ so $$Y(f)=\boxed{2\delta(f)+\big[\delta(f-f_0)+\delta(f+f_0)\big]+\big[\delta(f-2f_0)+\delta(f+2f_0)\big]}$$ — a DC term plus the original tone plus a NEW second-harmonic tone, exactly the signature of the input$^2$ nonlinearity (confirmed: $x(t)+x(t)^2$ evaluated numerically matches this closed form to $10^{-9}$ over $t\in[0,1]$).
2. Part (b) — Z(f), z(t). Evaluate $H(f)$ at each of the three frequencies present in $Y(f)$:
   • $f=0$: inside the flat passband ($|H|=1$) and the phase curve itself passes through $0$ there ($\angle H(0)=0$).
   • $f=\pm f_0$: since $f_0<1.1f_0$, still inside the flat passband, $|H(f_0)|=1$; and since $f_0>0.5f_0$, this sits in the region where the phase is flat at $0$, so $\angle H(f_0)=0$ too.
   • $f=\pm2f_0$: since $2f_0>1.5f_0$, this is fully in the stopband, $|H(2f_0)|=0$.
   So the DC and $f_0$ components pass through $H(f)$ completely unchanged (unit gain, zero phase), while the $2f_0$ component is entirely rejected: $$Z(f)=\boxed{2\delta(f)+\delta(f-f_0)+\delta(f+f_0)},\quad z(t)=\boxed{2+2\cos(2\pi f_0t)}.$$
3. Part (c) — linear? distortionless? Compare $z(t)=2+2\cos(2\pi f_0t)$ with $x(t)=2\cos(2\pi f_0t)$.
   Linearity: $z(t)$ contains a DC (zero-frequency) component that is completely ABSENT from $x(t)$. An LTI system is an eigenfunction system for sinusoids — a single-frequency input can only produce output at that SAME frequency (scaled/phase-shifted), never a new one. The appearance of a new frequency component (here, DC) from a purely single-tone input is proof by itself that the cascade in the dashed box is NOT linear.
   Distortionless: a distortionless system must satisfy $z(t)=K\,x(t-t_d)$ for constants $K,t_d$ — i.e. the SAME waveform shape, only scaled and delayed. $x(t)$ has zero average value while $z(t)$ has nonzero average value $2$; no choice of $K,t_d$ can turn a zero-mean cosine into a shifted-mean one, so the cascade is NOT distortionless either — consistent with (and implied by) it already being nonlinear.
4. Part (d) — black-box test design. Linearity test: drive the box with two SEPARATE, unequal-frequency sinusoids $x_1(t)$ and $x_2(t)$, record each output, then drive with $x_1(t)+x_2(t)$ and compare the recorded sum to the actual combined-input output (the superposition test) — any mismatch proves nonlinearity, and it is the only fully conclusive test. A cheaper NECESSARY (not sufficient) check: drive with a single sinusoid at $f_1$ and look at the output spectrum — any energy at a frequency other than $f_1$ (a harmonic, or DC as found here) proves the box is nonlinear; seeing energy ONLY at $f_1$ does not by itself prove linearity without the full superposition test.
   Distortionless test: sweep a single-tone input across the signal band of interest and measure, at each frequency, the output/input magnitude ratio and phase difference. Distortionless transmission requires the magnitude ratio to be CONSTANT across the band (flat amplitude response) AND the phase to be LINEAR in frequency (equivalently, constant group delay $-\tfrac1{2\pi}\tfrac{d\angle H}{df}$). A two-tone test is a useful cross-check: distortionless transmission must preserve both the relative amplitude ratio and the relative phase difference between the two tones end to end.