Q2Digital Signal Processing

(a) Two-Fold Expanded Signal

Note on the given problem statement: the input signal is specified as x(n) = x for x>0, 0 otherwise - this notational form is somewhat ambiguous/unusual as literally written (using 'x' as both the sequence name and an apparent placeholder condition variable), and most plausibly represents an incompletely-transcribed reference to a specific example sequence rather than a fully general formula. Rather than force an uncertain interpretation of this specific ambiguous notation, the general two-fold expansion (interpolation zero-insertion with L=2) procedure is demonstrated clearly below using a concrete illustrative example sequence, which directly demonstrates the requested operation.

General two-fold expansion procedure: two-fold expansion (zero-insertion by L=2) of an input sequence x(n) produces an output y(n) defined as y(n)=x(n/2) when n is even (i.e., n/2 is a valid integer index into the original sequence), and y(n)=0 when n is odd, effectively inserting exactly one zero-valued sample between each pair of consecutive original samples.

Illustrative example: suppose the input sequence is x(n) = {3, 5, 2, 4} for n=0,1,2,3 (a small, concrete example sequence chosen to clearly demonstrate the expansion procedure). The two-fold expanded output is:

where y(0)=x(0)=3, y(1)=0 (inserted zero), y(2)=x(1)=5, y(3)=0 (inserted zero), y(4)=x(2)=2, y(5)=0 (inserted zero), y(6)=x(3)=4, y(7)=0 (inserted zero) - directly demonstrating the general zero-insertion pattern. This zero-stuffed signal, when subsequently passed through a low-pass anti-imaging (interpolation) filter (as discussed in the companion decimation/interpolation answer elsewhere in this paper), would produce a fully-interpolated version of the original sequence at twice the original sampling rate, with the zero-valued samples smoothly filled in by meaningfully-interpolated intermediate values rather than remaining as literal zeros.

(b) Discrete-Time Processing of Continuous-Time Signals

Discrete-time processing of continuous-time signals refers to the reverse configuration of the continuous-time-processing-of-discrete-time-signals technique (discussed in the companion 2021 paper's u1-q1(b) answer): here, a continuous-time input signal xc(t) is first converted to discrete-time via an ideal C/D (continuous-to-discrete, i.e., sampling) converter, producing x(n)=xc(nT); this discrete-time sequence is then processed by a discrete-time (digital) LTI system with frequency response H(e^jw), producing a discrete-time output y(n); finally, this output is converted back to continuous-time via an ideal D/C (discrete-to-continuous, i.e., reconstruction) converter, producing the final continuous-time output yc(t).

Equivalent continuous-time system: provided xc(t) is adequately band-limited relative to the chosen sampling period T (satisfying the sampling theorem), this C/D-digital-processing-D/C chain behaves exactly as an equivalent continuous-time LTI system with frequency response Hc(jOmega) related to the discrete-time system's frequency response by Hc(jOmega) = H(e^(jOmegaT)) for |Omega| < pi/T (within the Nyquist frequency range corresponding to the chosen sampling rate). This equivalence is the fundamental principle underlying the entire modern practice of digital signal processing applied to real-world, continuous-time signals (audio, communications, sensor signals, and so on) - by first sampling the continuous signal (with an appropriate anti-aliasing filter to ensure adequate band-limiting), all the actual processing (filtering, transformation, analysis) can be performed entirely digitally, using flexible, precise, and reconfigurable digital hardware/software, before finally reconstructing a continuous-time output signal, achieving performance and flexibility advantages over purely analog processing throughout the vast majority of modern signal-processing applications.

The two-fold expansion operation (up-sampling by a factor of 2) considered here is the zero-insertion step that precedes the anti-imaging low-pass filter in a complete interpolation-by-2 system, and it is important to distinguish this zero-insertion expansion from actual interpolation: the expanded sequence y(n) = x(n/2) for even n, and y(n)=0 for odd n, by itself does not yet represent a faithful higher-rate reconstruction of the original continuous-time signal, since the inserted zero samples are simply placeholders rather than genuine interpolated signal values - a low-pass filter with cutoff at pi/2 (half the new, expanded sampling rate's Nyquist frequency) must be applied to y(n) afterward to replace those zero placeholders with properly interpolated values and to remove the spectral images introduced by the zero-insertion process, completing the true interpolation-by-2 operation.

Discrete-time processing of continuous-time signals is the foundational paradigm underlying essentially all modern digital signal processing of real-world analog signals, and it proceeds through three conceptual stages: first, an analog-to-digital converter (preceded by an analog anti-aliasing filter satisfying the sampling theorem's Nyquist criterion) converts the continuous-time signal xa(t) into a discrete-time sequence x(n)=xa(nT); second, the discrete-time sequence is processed by a digital system (implementing, for example, a digital filter, a spectral analysis operation, or any other discrete-time algorithm) to produce an output sequence y(n); and third, a digital-to-analog converter (typically followed by an analog reconstruction/smoothing filter) converts y(n) back into a continuous-time output signal ya(t).

The overall continuous-time-equivalent system realized by this three-stage discrete-time-processing chain can, under the idealized condition that the sampling rate exceeds the Nyquist rate for both the input and the processing bandwidth of interest, be shown to behave as an equivalent continuous-time LTI system whose frequency response is directly related to the discrete-time system's frequency response by the simple frequency-scaling relationship Omega = w/T - this equivalence is what allows digital filters to be designed, analyzed, and specified directly in terms of desired continuous-time (analog) frequency-response specifications, and is the theoretical justification underlying frequency-transformation-based digital filter design techniques such as impulse invariance and the bilinear transformation used extensively elsewhere in this examination paper.

It is also worth noting the frequency-domain picture of this two-fold expansion operation explicitly: if X(e^jw) denotes the DTFT of the original sequence x(n), the DTFT of the zero-inserted (expanded) sequence y(n) is given by Y(e^jw) = X(e^j2w), a frequency-axis compression by the up-sampling factor of 2 - this compression is exactly what produces the additional spectral image (a compressed copy of the original spectrum appearing within the expanded sequence's Nyquist range) that the subsequent low-pass interpolation filter, with cutoff at pi/2, is specifically designed to remove, leaving only the correctly interpolated baseband copy of the original signal's spectrum in the final output.

This three-stage discrete-time-processing-of-continuous-time-signals paradigm also clarifies the practical role of the two analog filters (anti-aliasing before the ADC, and reconstruction/smoothing after the DAC) that bracket the digital processing core: the anti-aliasing filter ensures the input signal satisfies the sampling theorem's bandwidth condition before sampling, preventing irreversible aliasing distortion, while the reconstruction filter smooths the DAC's inherently stepped (zero-order-hold) output waveform back into a genuinely continuous signal, removing the spurious high-frequency spectral images introduced by the zero-order-hold reconstruction process - both analog filters are necessary complements to the digital processing core, and their design requirements (cutoff frequency, roll-off steepness) are directly determined by the chosen sampling rate relative to the signal bandwidth of interest.