Ideal Reconstruction with Sinc Interpolation

Hello! Welcome to our next lesson on the journey from analog to digital signals.

In our last session, we investigated the critical phenomenon of aliasing. We established that if a signal $x (t)$ is band-limited to $B$ Hz, sampling at a rate $f_{s} > 2 B$ ensures that the spectral replicas in the sampled signal's spectrum, $X_{s} (f)$ , do not overlap. This non-overlapping condition is the key that unlocks the possibility of perfectly recovering the original continuous-time signal from its discrete samples.

Today, we move from possibility to practice. Our goal is to derive the exact mathematical procedure for this perfect recovery. We will show how the ideal low-pass filtering we conceptualized in the frequency domain translates into a concrete time-domain operation, leading us to one of the most elegant results in signal processing: the Whittaker-Shannon interpolation formula.

1. The Path to Reconstruction: From Frequency to Time

Let's start where we left off. We have an impulse-sampled signal, $x_{s} (t)$ , whose Fourier transform is a series of scaled and shifted replicas of the original spectrum $X (ω)$ :

$X_{s} (ω) = \frac{1}{T _{s}} \sum_{k = - \infty}^{\infty} X (ω - k ω_{s})$

Since we sampled above the Nyquist rate ( $ω_{s} > 2 ω_{m}$ ), the replicas are distinct. To recover the original spectrum, $X (ω)$ , we need to isolate the central replica (the one for $k = 0$ ) and discard all others.

This is a perfect job for a filter. Specifically, an ideal low-pass filter. This filter would:

Pass all frequencies in the original signal's band, $∣ ω ∣ < ω_{m}$ .
Completely block all frequencies beyond that, where the unwanted replicas lie.
Compensate for the $1/ T_{s}$ scaling factor introduced by the sampling process.

The frequency response of such an ideal reconstruction filter, $H_{r} (ω)$ , is a rectangular pulse:

H_{r} (ω) = {T_{s} 0 ∣ ω ∣ < ω_{s} /2 ∣ ω ∣ \geq ω_{s} /2

where we use a cutoff of $ω_{s} /2$ to be safe. By applying this filter, we get back our original spectrum:
$X (ω) = X_{s} (ω) \cdot H_{r} (ω)$

This is clear in the frequency domain. But how do we perform this operation in the time domain? Recalling the convolution theorem, we know that multiplication in the frequency domain is equivalent to convolution in the time domain. This gives us our reconstruction equation:

$x (t) = x_{s} (t) * h_{r} (t)$

Here, $x_{s} (t)$ is our impulse-sampled signal, and $h_{r} (t)$ is the impulse response of our ideal reconstruction filter—the inverse Fourier transform of $H_{r} (ω)$ . Our next task is to find this impulse response.

2. Deriving the Ideal Reconstruction Formula

Finding $h_{r} (t)$ and performing the convolution with the impulse train $x_{s} (t)$ are the two key steps to deriving the final formula. The following video provides a clear, step-by-step derivation that aligns perfectly with the mathematical rigor you appreciate.

Sampling Signals (11/13) - Ideal Band-Limited Interpolation (i.e. Sinc Interpolation)

This video by Adam Panagos, titled 'Ideal Band-Limited Interpolation', will walk you through the entire derivation. It starts with the frequency-domain filtering concept we just discussed and meticulously derives the time-domain equivalent.

Please watch from 00:57 to 07:05. Focus on two main parts: Finding the Impulse Response (02:13 - 05:40): Pay close attention to how the inverse Fourier transform of the rectangular frequency response H_r(\omega) is found. Note the use of the standard Fourier transform pair for a rectangular pulse and how the scaling factors simplify to 1, resulting in a clean sinc function for the impulse response h_r(t). Convolution with the Impulse Train (05:40 - 07:05): Observe how the convolution x_s(t) * h_r(t) is resolved. The video demonstrates how convolving a function with a train of weighted impulses simplifies to a summation of shifted and weighted versions of that function.

Let's summarize the elegant result from the video.

Step 1: Find the Impulse Response $h_{r} (t)$

The impulse response is the inverse Fourier transform of the rectangular frequency response $H_{r} (ω)$ . As the video demonstrates, a rectangular pulse in one domain corresponds to a sinc function in the other. After accounting for all scaling factors, the impulse response of the ideal reconstruction filter is:

$h_{r} (t) = F^{- 1} {H_{r} (ω)} = sinc (\frac{ω _{s} t}{2}) = sinc (\frac{π t}{T _{s}})$
(Note: Here we use the normalized sinc function, $sinc (x) = sin (π x) / (π x)$ . The video uses $sinc (x) = sin (x) / x$ , which is also common; the key is consistency. The final formula's structure is the same.)

Step 2: Perform the Convolution

The reconstructed signal is the convolution of the impulse-sampled signal with this sinc function.
$x (t) = x_{s} (t) * h_{r} (t) = (\sum_{n = - \infty}^{\infty} x (n T_{s}) δ (t - n T_{s})) * h_{r} (t)$
Using the linearity of convolution and the sifting property of the delta function ( $δ (t - T) * f (t) = f (t - T)$ ), we get:
$x (t) = \sum_{n = - \infty}^{\infty} x (n T_{s}) (δ (t - n T_{s}) * h_{r} (t)) = \sum_{n = - \infty}^{\infty} x (n T_{s}) h_{r} (t - n T_{s})$

The Final Formula

Substituting the sinc function for $h_{r} (t)$ , we arrive at the celebrated Whittaker-Shannon Interpolation Formula:

$x (t) = \sum_{n = - \infty}^{\infty} x (n T_{s}) \cdot sinc (\frac{t - n T _{s}}{T _{s}})$

This equation is the mathematical heart of ideal signal reconstruction. It tells us precisely how to rebuild the original continuous-time function $x (t)$ from its discrete samples $x (n T_{s})$ .

3. Visualizing Sinc Interpolation

What does this formula mean intuitively? It states that the original signal can be reconstructed by placing a sinc function at the location of each sample, scaling its amplitude by the sample's value, and then summing all of these infinite sinc functions together.

Ideal Bandlimited Interpolation and Sinc Reconstruction — This diagram perfectly illustrates the reconstruction process. The original continuous signal (solid line) is reconstructed by summing an infinite series of sinc functions (dashed lines). Each sinc is centered at a sample time $nT_s$ and scaled by the sample value $x(nT_s)$. Notice how the peak of each sinc corresponds to a sample point, and how it is zero at all other sample points.

The following short reading provides another concise derivation and a clear graphical representation, reinforcing this concept.

EE 424 #1: Sampling and Reconstruction

These lecture notes from Iowa State University, 'EE 424 #1: Sampling and Reconstruction', offer a compact summary of the Shannon interpolation formula and its graphical meaning.

Please read the section 'Ideal Reconstruction: Shannon interpolation formula' on page 13. Pay attention to how the convolution is expressed and how it leads to the final summation formula. Also, study Figure 13 on page 14, which reinforces the graphical interpretation we just discussed.

Test your understanding!

The sinc function, $sinc (\frac{t - n T _{s}}{T _{s}})$ , has a crucial property that makes this interpolation work. What is the value of this function at the time instant $t = k T_{s}$ , where $k$ is an integer? Consider two cases: $k = n$ and $k \neq = n$ . How does this property ensure that the reconstructed signal $x (t)$ passes exactly through the original sample points?

Show answer

The argument of the sinc function is $\frac{k T _{s} - n T _{s}}{T _{s}} = k - n$ .
Using the normalized sinc, $sinc (x) = sin (π x) / (π x)$ .

Case 1: $k = n$
The argument is $k - n = 0$ . The value is $sinc (0) = 1$ .
Case 2: $k \neq = n$
The argument $k - n$ is a non-zero integer. The value is $sinc (integer) = sin (π \cdot integer) / (π \cdot integer) = 0$ .

This means that at any given sample instant $t = k T_{s}$ , only one term in the infinite summation is non-zero: the term where $n = k$ . All other sinc functions in the sum are exactly zero at this point. The value of the sum at $t = k T_{s}$ is therefore:
$x (k T_{s}) = \sum_{n = - \infty}^{\infty} x (n T_{s}) \cdot δ_{nk} = x (k T_{s}) \cdot 1 = x (k T_{s})$
This proves that the reconstructed signal perfectly agrees with the original sample values at every sampling instant.

4. The Catch: Why "Ideal"?

The Whittaker-Shannon formula is mathematically perfect, but it relies on an "ideal" filter. In engineering, "ideal" is often a synonym for "unrealizable." There are two main reasons for this:

Non-causality: The sinc function $h_{r} (t)$ is non-zero for $t < 0$ . A causal system's impulse response must be zero for negative time. To generate the output at time $t$ , this filter would need to know the input samples from the future ( $n > t / T_{s}$ ). This is impossible for a real-time system.
Infinite Duration: The sinc function extends from $t = - \infty$ to $t = + \infty$ . Implementing it would require an infinitely long filter and an infinite number of samples, which is not practical.

This is the fundamental trade-off. The "brick-wall" rectangular filter in the frequency domain, which provides perfect spectral separation, corresponds to the non-causal, infinite-duration sinc function in the time domain.

Conclusion

In this lesson, we bridged the gap between the theory of sampling and the mathematics of reconstruction. We derived the mechanism for perfect signal recovery from its samples, assuming the Nyquist criterion is met.

Key Takeaways:

Ideal reconstruction is achieved by applying an ideal low-pass filter to an impulse-sampled signal.
The impulse response of this ideal filter is the sinc function.
In the time domain, this filtering operation becomes a convolution, which resolves to the Whittaker-Shannon interpolation formula:
$x (t) = \sum_{n = - \infty}^{\infty} x (n T_{s}) \cdot sinc (\frac{t - n T _{s}}{T _{s}})$
This formula reconstructs the continuous signal by summing an infinite series of scaled and time-shifted sinc functions.
The reconstruction is "ideal" because the required sinc filter is non-causal and has an infinite impulse response, making it physically unrealizable.

Preview of the Next Lesson:
We've established the theoretical benchmark for reconstruction. But if the ideal filter is impossible to build, how do we reconstruct signals in the real world? In our next lesson, we will address this critical question by exploring the practical necessity and design considerations for anti-aliasing and reconstruction filters. We will see how real-world filters approximate the ideal and what trade-offs are involved.

Recommended resources

1. The Path to Reconstruction: From Frequency to Time

2. Deriving the Ideal Reconstruction Formula

3. Visualizing Sinc Interpolation

4. The Catch: Why "Ideal"?

Conclusion