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Huiswerkopgaven 5a - Conceptual questions

Q: The Fourier transform of the impulse train p(t) with period Ts is itself an impulse train in the frequency domain: Using the convolution theorem, write an expression for Gs(ω), the Fourier transform of the sampled signal

Because multiplication in time corresponds to convolution in frequency, The sampled spectrum is a superposition of shifted copies of G(ω), spaced ωs apart and scaled by 1/Ts.

Q: Define aliasing. In what sense does an aliased frequency component “masquerade” as a different frequency?

Aliasing occurs when a signal is sampled below the Nyquist rate, so that spectral copies of the original overlap in the sampled spectrum. High-frequency components “fold” into the baseband and are indistinguishable from lower-frequency signals; after reconstruction they appear at the wrong (alias) frequency

Q: Give one real-world example of aliasing (not from audio) and briefly explain what the alias looks like.

A common example is the “wagon wheel” effect in film: spokes rotating faster than the camera frame rate appear to rotate slowly backwards, because the camera samples the scene at a fixed frame rate and high rotation frequencies alias to low (or even negative) frequencies

Q: What is an anti-aliasing filter? Where in the signal chain is it placed, and what type of filter is it?

An anti-aliasing filter is an analogue low-pass filter placed before the ADC (analogue-to-digital converter). It attenuates all frequency components above fs/2 (the Nyquist frequency) so that, after sampling, no out-of-band content can alias into the baseband

Q: Does applying an anti-aliasing filter distort the signal? Explain what information is lost and why engineers accept this trade-of.

Yes, the filter removes genuine high-frequency content that was present in the original signal. That information is permanently lost. Engineers accept this because the alternative – aliased distortion – is usually far more objectionable: it introduces spurious tones that were never in the original signal, whereas the filtered signal is simply band-limited

Q: Suppose a microphone captures audio up to 20 kHz and the system will sample at 44.1 kHz. What should the cut-off frequency of the anti-aliasing filter be, and why?

The Nyquist frequency is fs/2 = 22.05 kHz. The anti-aliasing filter cut-off must be at most 22.05 kHz to prevent aliasing. Since the full audible range extends only to 20 kHz, a standard choice is fc = 20 kHz: this preserves the entire audible band while ensuring nothing above 22.05 kHz folds back into the baseband. (The small gap 20–22.05 kHz gives the filter’s roll-off region room to operate.)

8 important questions on Huiswerkopgaven 5a - Conceptual questions

State the Nyquist–Shannon sampling theorem precisely, defining all terms you use (sampling rate, Nyquist rate, bandwidth)

The Nyquist–Shannon sampling theorem states: a band-limited signal whose highest frequency component is fm can be perfectly reconstructed from its samples if the sampling frequency satisfies fs ≥ 2fm. The value fN = 2fm is called the Nyquist rate; fm is the (one/sided) bandwidth of the signal.

A continuous-time signal g(t) is sampled by multiplying it by an impulse train

Write an expression for the sampled signal gs(t) and explain, in words, what information is retained in the samples and why choosing Ts too large causes a problem

Each impulse carries the instantaneous value g(nTs), so all information about g between samples is discarded. If Ts is too large (i.e. fs < 2fm), then not enough samples are taken per cycle of the highest-frequency component, and those high-frequency oscillations become indistinguishable from lower-frequency ones – the aliasing phenomenon.

The Fourier transform of the impulse train p(t) with period Ts is itself an impulse
train in the frequency domain:

Using the convolution theorem, write an expression for Gs(ω), the Fourier transform
of the sampled signal

Because multiplication in time corresponds to convolution in frequency, The sampled spectrum is a superposition of shifted copies of G(ω), spaced ωs
apart and scaled by 1/Ts.

Define aliasing. In what sense does an aliased frequency component “masquerade” as a different frequency?

Aliasing occurs when a signal is sampled below the Nyquist rate, so that spectral copies of the original overlap in the sampled spectrum. High-frequency components “fold” into the baseband and are indistinguishable from lower-frequency signals; after reconstruction they appear at the wrong (alias) frequency

Give one real-world example of aliasing (not from audio) and briefly explain what the alias looks like.

A common example is the “wagon wheel” effect in film: spokes rotating faster than the camera frame rate appear to rotate slowly backwards, because the camera samples the scene at a fixed frame rate and high rotation frequencies alias to low (or even negative) frequencies

What is an anti-aliasing filter? Where in the signal chain is it placed, and what type of filter is it?

An anti-aliasing filter is an analogue low-pass filter placed before the ADC (analogue-to-digital converter). It attenuates all frequency components above fs/2 (the Nyquist frequency) so that, after sampling, no out-of-band content can alias into the baseband

Does applying an anti-aliasing filter distort the signal? Explain what information is lost and why engineers accept this trade-of.

Yes, the filter removes genuine high-frequency content that was present in the original signal. That information is permanently lost. Engineers accept this because the alternative – aliased distortion – is usually far more objectionable: it introduces spurious tones that were never in the original signal, whereas the filtered signal is simply band-limited

Suppose a microphone captures audio up to 20 kHz and the system will sample at 44.1 kHz. What should the cut-off frequency of the anti-aliasing filter be, and why?

The Nyquist frequency is fs/2 = 22.05 kHz. The anti-aliasing filter cut-off must be at most 22.05 kHz to prevent aliasing. Since the full audible range extends only to 20 kHz, a standard choice is fc = 20 kHz: this preserves the entire audible band while ensuring nothing above 22.05 kHz folds back into the baseband. (The small gap 20–22.05 kHz gives the filter’s roll-off region room to operate.)

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Telecommunicatie

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Huiswerkopgaven 5a - Conceptual questions

8 important questions on Huiswerkopgaven 5a - Conceptual questions

State the Nyquist–Shannon sampling theorem precisely, defining all terms you use (sampling rate, Nyquist rate, bandwidth)

A continuous-time signal g(t) is sampled by multiplying it by an impulse trainWrite an expression for the sampled signal gs(t) and explain, in words, what information is retained in the samples and why choosing Ts too large causes a problem

The Fourier transform of the impulse train p(t) with period Ts is itself an impulsetrain in the frequency domain:Using the convolution theorem, write an expression for Gs(ω), the Fourier transformof the sampled signal

Define aliasing. In what sense does an aliased frequency component “masquerade” as a different frequency?

Give one real-world example of aliasing (not from audio) and briefly explain what the alias looks like.

What is an anti-aliasing filter? Where in the signal chain is it placed, and what type of filter is it?

Does applying an anti-aliasing filter distort the signal? Explain what information is lost and why engineers accept this trade-of.

Suppose a microphone captures audio up to 20 kHz and the system will sample at 44.1 kHz. What should the cut-off frequency of the anti-aliasing filter be, and why?

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A continuous-time signal g(t) is sampled by multiplying it by an impulse train

Write an expression for the sampled signal gs(t) and explain, in words, what information is retained in the samples and why choosing Ts too large causes a problem

The Fourier transform of the impulse train p(t) with period Ts is itself an impulse
train in the frequency domain:

Using the convolution theorem, write an expression for Gs(ω), the Fourier transform
of the sampled signal