Signals and Systems

5.4 The Convolution Property

Imron Rosyadi

5.4 The Convolution Property

Today, we’re diving into one of the most important properties of the Discrete-Time Fourier Transform: the convolution property. This property is absolutely central to the analysis and design of discrete-time Linear Time-Invariant (LTI) systems. It simplifies complex time-domain convolution operations into straightforward multiplications in the frequency domain, providing powerful insights into system behavior.

The Convolution Property

For an LTI system, if \(x[n]\) is the input, \(h[n]\) is the impulse response, and \(y[n]\) is the output:

\[ y[n]=x[n] * h[n] \]

Then, their Discrete-Time Fourier Transforms are related by:

\[ Y\left(e^{j \omega}\right)=X\left(e^{j \omega}\right) H\left(e^{j \omega}\right) \quad \text{(Eq. 5.48)} \]

• Convolution in the time domain becomes multiplication in the frequency domain.
• \(H(e^{j\omega})\) is the frequency response of the LTI system, which is the DTFT of the impulse response \(h[n]\).

Important

This property is a cornerstone of LTI system analysis, allowing us to analyze system behavior in the frequency domain, which is often much simpler.

This equation is incredibly powerful. It tells us that the complex operation of convolution in the time domain, which can be quite tedious to compute, transforms into a simple multiplication in the frequency domain. This is why Fourier transforms are so fundamental to LTI system analysis. The term \(H(e^{j\omega})\) is what we call the frequency response, and it essentially tells us how the system modifies the amplitude and phase of each frequency component of the input signal.

Why is it so useful?

• Simplifies Analysis: Converts complex convolution into algebraic multiplication.
• System Understanding: \(H(e^{j\omega})\) directly shows how a system affects different frequencies.
  • Magnitude Response \(|H(e^{j\omega})|\): Gain at each frequency.
  • Phase Response \(\operatorname{Arg}\{H(e^{j\omega})\}\): Phase shift at each frequency.
• Filter Design: Allows direct specification of desired frequency characteristics (e.g., passband, stopband) to design filters.
• Eigenfunction Interpretation: Complex exponentials \(e^{j\omega n}\) are eigenfunctions of LTI systems. The frequency response \(H(e^{j\omega})\) is the eigenvalue for \(e^{j\omega n}\).

The usefulness of the convolution property cannot be overstated. It drastically simplifies the analysis of LTI systems. Instead of performing lengthy convolutions, we can work with simpler multiplications. The frequency response \(H(e^{j\omega})\) becomes a direct window into how the system processes signals. We can easily see which frequencies are amplified or attenuated (magnitude response) and how their phases are shifted (phase response). This is the basis for filter design, where we specify a desired frequency response and then find an impulse response that achieves it. It also ties back to the concept of complex exponentials being eigenfunctions of LTI systems, with \(H(e^{j\omega})\) being the corresponding eigenvalue.

Example 5.11: Time Shift System

Consider an LTI system with impulse response:

\[ h[n]=\delta\left[n-n_{0}\right] \]

This system simply delays the input by \(n_0\) samples: \(y[n] = x[n-n_0]\).

1. Find the frequency response \(H(e^{j\omega})\):

\[ H\left(e^{j \omega}\right)=\mathcal{F}\{\delta[n-n_0]\} = \sum_{n=-\infty}^{+\infty} \delta\left[n-n_{0}\right] e^{-j \omega n}=e^{-j \omega n_{0}} \]

2. Apply the convolution property:

\[ Y\left(e^{j \omega}\right)=X\left(e^{j \omega}\right) H\left(e^{j \omega}\right)=X\left(e^{j \omega}\right) e^{-j \omega n_{0}} \quad \text{(Eq. 5.49)} \]

This is consistent with the time-shifting property: \(x[n-n_0] \stackrel{\mathcal{F}}{\longleftrightarrow} e^{-j\omega n_0} X(e^{j\omega})\).

Note

The frequency response \(H(e^{j\omega})=e^{-j\omega n_0}\) has:

• Unity magnitude: \(|H(e^{j\omega})|=1\) for all \(\omega\). (No frequency is amplified or attenuated)
• Linear phase: \(\operatorname{Arg}\{H(e^{j\omega})\} = -\omega n_0\). (All frequencies are delayed by the same amount, \(n_0\) samples, resulting in a constant group delay).

Let’s start with a simple LTI system: a pure time delay. Its impulse response is a shifted impulse. First, we find its frequency response by taking the DTFT of the impulse response, which gives us \(e^{-j\omega n_0}\). Then, applying the convolution property, the output spectrum is simply the input spectrum multiplied by this phase term. This result perfectly aligns with the time-shifting property we discussed earlier. It clearly shows that a pure delay system does not change the magnitude of any frequency component, but it introduces a linear phase shift, meaning all frequencies are delayed by the same amount. This is an ideal delay, often desired in systems.

Example 5.12: Ideal Discrete-Time Lowpass Filter

An ideal lowpass filter has a frequency response \(H(e^{j\omega})\) as shown:

Figure 5.17(a): Ideal lowpass filter frequency response.

We can find its impulse response \(h[n]\) using the inverse DTFT:

\[ \begin{aligned} h[n] & =\frac{1}{2 \pi} \int_{-\pi}^{\pi} H\left(e^{j \omega}\right) e^{j \omega n} d \omega \\ & =\frac{1}{2 \pi} \int_{-\omega_{c}}^{\omega_{c}} e^{j \omega n} d \omega \\ & =\frac{\sin \omega_{c} n}{\pi n} \quad \text{(Eq. 5.50)} \end{aligned} \]

Figure 5.17(b): Impulse response of the ideal lowpass filter.

Warning

Challenges with Ideal Filters:

• Non-causal: \(h[n]\) is non-zero for \(n<0\). Ideal filters cannot be implemented in real-time.
• Oscillatory: The impulse response is an oscillating sinc function, which can lead to undesirable time-domain ringing artifacts.
• Trade-offs: Practical filter design involves trade-offs between ideal frequency response and realizable time-domain characteristics.

The ideal lowpass filter is a theoretical construct, but it’s very useful for understanding filter design. Its frequency response is a rectangle in the frequency domain. To find its impulse response, we perform the inverse DTFT. This gives us a sinc function. This impulse response immediately highlights some practical challenges. Since \(h[n]\) is non-zero for negative \(n\), the filter is non-causal, meaning it would require future inputs to compute the current output, which is impossible in real-time. Also, the oscillatory nature of the sinc function can cause “ringing” artifacts in the output, which might be undesirable. These are important trade-offs we face when designing practical filters.

Interactive Plot: Ideal Lowpass Filter \(h[n]\)

Observe the impulse response \(h[n]\) of an ideal lowpass filter for different cutoff frequencies.

viewof wc_val = Inputs.range([0.1, 3.14 - 0.1], {value: 3.14/2, step: 0.05, label: "Cutoff Frequency $\omega_c$"});

This interactive plot allows you to change the cutoff frequency \(\omega_c\) of an ideal lowpass filter. As you increase \(\omega_c\), the main lobe of the sinc function becomes narrower, and its peak value increases. This means the filter becomes “sharper” in frequency but more concentrated around \(n=0\) in time. Conversely, a smaller \(\omega_c\) makes the sinc wider and lower. Notice how the impulse response extends infinitely in both positive and negative \(n\), confirming its non-causal nature. This visualization helps connect the frequency-domain design of an ideal filter to its time-domain characteristics.

Example 5.13: Convolution of Exponentials

Let’s use the convolution property to find the output \(y[n]\) of an LTI system.

• Impulse response: \(h[n]=\alpha^{n} u[n]\), with \(|\alpha|<1\).
• Input signal: \(x[n]=\beta^{n} u[n]\), with \(|\beta|<1\).

1. Find DTFTs of \(h[n]\) and \(x[n]\) (from Example 5.1):

\[ H\left(e^{j \omega}\right)=\frac{1}{1-\alpha e^{-j \omega}} \quad \text{(Eq. 5.51)} \]

\[ X\left(e^{j \omega}\right)=\frac{1}{1-\beta e^{-j \omega}} \quad \text{(Eq. 5.52)} \]

2. Multiply in frequency domain:

\[ Y\left(e^{j \omega}\right)=H\left(e^{j \omega}\right) X\left(e^{j \omega}\right)=\frac{1}{\left(1-\alpha e^{-j \omega}\right)\left(1-\beta e^{-j \omega}\right)} \quad \text{(Eq. 5.53)} \]

3. Find inverse DTFT of \(Y(e^{j\omega})\) using partial fractions:

• Case 1: \(\alpha \neq \beta\) \[ Y\left(e^{j \omega}\right)=\frac{A}{1-\alpha e^{-j \omega}}+\frac{B}{1-\beta e^{-j \omega}} \] where \(A=\frac{\alpha}{\alpha-\beta}\) and \(B=-\frac{\beta}{\alpha-\beta}\). Taking the inverse DTFT (using linearity and Example 5.1): \[ y[n]=\frac{1}{\alpha-\beta}\left[\alpha^{n+1} u[n]-\beta^{n+1} u[n]\right] \quad \text{(Eq. 5.55)} \]

• Case 2: \(\alpha = \beta\) \[ Y\left(e^{j \omega}\right)=\left(\frac{1}{1-\alpha e^{-j \omega}}\right)^{2} \] Using the differentiation in frequency property (Eq. 5.46) and time-shifting property: \[ y[n]=(n+1) \alpha^{n} u[n] \quad \text{(Eq. 5.58)} \]

This example showcases a common application of the convolution property: calculating a convolution sum. Direct convolution of two exponentials can be tedious. However, by transforming them to the frequency domain, we perform a simple multiplication. The challenge then shifts to finding the inverse DTFT of the product. This is where techniques like partial fraction expansion become invaluable. We consider two cases: when the exponential bases are different, and when they are the same. For the latter, the differentiation in frequency property comes in handy, illustrating how different DTFT properties often work together to solve problems.

Example 5.14: System Interconnection Analysis

Consider the system shown in Figure 5.18(a).

Figure 5.18(a): System interconnection.

• \(H_{lp}(e^{j\omega})\) are ideal lowpass filters with cutoff \(\pi/4\).
• The “\(-1\)” blocks multiply by \((-1)^n = e^{j\pi n}\).

Example 5.14: System Interconnection Analysis

Analysis of Paths:

• Top Path:

  1. \(w_1[n] = (-1)^n x[n] \stackrel{\mathcal{F}}{\longleftrightarrow} W_1(e^{j\omega}) = X(e^{j(\omega-\pi)})\) (Frequency Shift)
  2. \(w_2[n] = w_1[n] * h_{lp}[n] \stackrel{\mathcal{F}}{\longleftrightarrow} W_2(e^{j\omega}) = H_{lp}(e^{j\omega}) X(e^{j(\omega-\pi)})\) (Convolution Property)
  3. \(w_3[n] = (-1)^n w_2[n] \stackrel{\mathcal{F}}{\longleftrightarrow} W_3(e^{j\omega}) = W_2(e^{j(\omega-\pi)}) = H_{lp}(e^{j(\omega-\pi)}) X(e^{j(\omega-2\pi)})\) Since \(X(e^{j(\omega-2\pi)}) = X(e^{j\omega})\) (DTFT Periodicity): \(W_3(e^{j\omega}) = H_{lp}(e^{j(\omega-\pi)}) X(e^{j\omega})\)

• Lower Path:

  \(W_4(e^{j\omega}) = H_{lp}(e^{j\omega}) X(e^{j\omega})\) (Convolution Property)

Example 5.14: System Interconnection Analysis

Overall System:

\[ Y\left(e^{j \omega}\right) = W_{3}\left(e^{j \omega}\right)+W_{4}\left(e^{j \omega}\right) = \left[H_{l p}\left(e^{j(\omega-\pi)}\right)+H_{l p}\left(e^{j \omega}\right)\right] X\left(e^{j \omega}\right) \]

The overall frequency response is \(H(e^{j\omega}) = H_{lp}(e^{j(\omega-\pi)}) + H_{lp}(e^{j\omega})\).

Figure 5.18(b): Overall frequency response.

Tip

\(H_{lp}(e^{j(\omega-\pi)})\) is the frequency response of an ideal highpass filter (from Example 5.7). Thus, the overall system creates an ideal bandstop filter, passing low and high frequencies, and stopping frequencies between \(\pi/4\) and \(3\pi/4\).

This complex system interconnection problem elegantly demonstrates the power of using multiple DTFT properties together. We break down the system into individual LTI blocks and operations. The “\(-1\)” blocks correspond to multiplication by \((-1)^n\), which we know causes a frequency shift by \(\pi\). The LTI filter blocks use the convolution property. By applying frequency shifting, convolution, and linearity properties sequentially, we can derive the overall frequency response of the cascaded and parallel system. The final result is a bandstop filter, which stops a specific band of frequencies while allowing others to pass. This kind of analysis is common in designing communication systems, audio processing, and other signal processing applications.

Stability and Frequency Response

• Not all LTI systems have a well-defined frequency response.
  • Example: \(h[n] = 2^n u[n]\) (unstable system). Its DTFT (analysis equation) diverges.
• For a stable LTI system, its impulse response \(h[n]\) must be absolutely summable:

\[ \sum_{n=-\infty}^{+\infty}|h[n]|<\infty \quad \text{(Eq. 5.59)} \]

• If an LTI system is stable, its frequency response \(H(e^{j\omega})\) always converges.
• For unstable systems, we use the z-transform (Chapter 10), which is a more general transform that includes the DTFT as a special case.

It’s crucial to remember that the frequency response, defined as the DTFT of the impulse response, only exists for stable LTI systems. A stable system is one whose impulse response is absolutely summable. If the system is unstable, like an exponential growing without bound, its DTFT will not converge. In such cases, we cannot use the frequency response to analyze the system. For these scenarios, we turn to a more generalized transform called the z-transform, which we will cover in a later chapter. For now, we will assume that the systems we analyze using the DTFT are stable.