Signals and Systems

5.3 Properties of the Discrete-Time Fourier Transform

Imron Rosyadi

Notation and Overview

We’ll use a concise notation to indicate Fourier transform pairs:

\[ x[n] \stackrel{\mathcal{F}}{\longleftrightarrow} X\left(e^{j \omega}\right) \]

or

\[ X\left(e^{j \omega}\right) = \mathcal{F}\{x[n]\} \quad \text{and} \quad x[n] = \mathcal{F}^{-1}\{X\left(e^{j \omega}\right)\} \]

1. Periodicity
2. Linearity
3. Time Shifting
4. Frequency Shifting
5. Conjugation & Conjugate Symmetry

6. Differencing & Accumulation
7. Time Reversal
8. Time Expansion
9. Differentiation in Frequency
10. Parseval’s Relation

This slide simply sets up the notation we’ll be using and gives you a roadmap of the properties we’re about to explore. This notation is standard and helps to quickly express the relationship between a time-domain signal and its frequency-domain transform.

1. Periodicity of the DTFT

The discrete-time Fourier transform \(X(e^{j\omega})\) is always periodic in \(\omega\) with period \(2\pi\).

\[ X\left(e^{j(\omega+2 \pi)}\right)=X\left(e^{j \omega}\right) \quad \text{(Eq. 5.28)} \]

Note

Contrast with CTFT:

The continuous-time Fourier transform is generally not periodic.

This periodicity in discrete time is a direct consequence of the nature of discrete-time complex exponentials \(e^{j\omega n}\), which are also periodic in \(\omega\) with period \(2\pi\).

This is a fundamental property we’ve already highlighted. Unlike its continuous-time counterpart, the DTFT always repeats itself every \(2\pi\) radians. This means we only need to analyze the spectrum over any \(2\pi\) interval, typically from \(-\pi\) to \(\pi\), or \(0\) to \(2\pi\). This periodicity is a direct result of the discrete nature of time.

2. Linearity of the Fourier Transform

The DTFT is a linear operation. If \(x_1[n] \stackrel{\mathcal{F}}{\longleftrightarrow} X_1(e^{j\omega})\) and \(x_2[n] \stackrel{\mathcal{F}}{\longleftrightarrow} X_2(e^{j\omega})\), then:

\[ a x_{1}[n]+b x_{2}[n] \stackrel{\mathcal{F}}{\longleftrightarrow} a X_{1}\left(e^{j \omega}\right)+b X_{2}\left(e^{j \omega}\right) \quad \text{(Eq. 5.29)} \]

• This means the Fourier transform of a sum of scaled signals is the sum of their scaled Fourier transforms.
• Extremely useful for breaking down complex signals into simpler components.

Linearity is a powerful property common to many transforms. It means that if you have a linear combination of signals in the time domain, their Fourier transform will be the same linear combination of their individual Fourier transforms. This allows us to analyze signals composed of multiple parts by finding the DTFT of each part separately and then combining them.

3. Time Shifting and 4. Frequency Shifting

If \(x[n] \stackrel{\mathcal{F}}{\longleftrightarrow} X(e^{j\omega})\):

Time Shifting:

Shifting a signal in time introduces a linear phase shift in its spectrum.

\[ x\left[n-n_{0}\right] \stackrel{\mathcal{F}}{\longleftrightarrow} e^{-j \omega n_{0}} X\left(e^{j \omega}\right) \quad \text{(Eq. 5.30)} \]

• A delay by \(n_0\) samples results in multiplication by \(e^{-j\omega n_0}\) in the frequency domain.

Frequency Shifting:

Multiplying a signal by a complex exponential in time shifts its spectrum in frequency.

\[ e^{j \omega_{0} n} x[n] \stackrel{\mathcal{F}}{\longleftrightarrow} X\left(e^{j\left(\omega-\omega_{0}\right)}\right) \quad \text{(Eq. 5.31)} \]

• This is the basis for modulation in communication systems.

These two properties are incredibly important. Time shifting tells us that if you delay a signal by \(n_0\) samples, its magnitude spectrum remains unchanged, but a linear phase term \(e^{-j\omega n_0}\) is introduced. This is crucial for understanding delays in systems. Conversely, frequency shifting states that multiplying a signal by a complex exponential in the time domain shifts its entire spectrum in the frequency domain. This is the fundamental principle behind modulation in communication systems, where information is shifted to different carrier frequencies.

Application: Ideal Highpass Filter from Lowpass

This property has a special application in filter design.

Lowpass Filter \(H_{lp}(e^{j\omega})\):

• Passes low frequencies.
• Cutoff frequency \(\omega_c\).

Figure 5.12(a): Lowpass filter frequency response.

Highpass Filter from Shift:

• Shifting \(H_{lp}(e^{j\omega})\) by \(\pi\): \(H_{hp}(e^{j\omega}) = H_{lp}(e^{j(\omega-\pi)})\) (Eq. 5.32)
• Since high frequencies are near \(\pi\), this effectively creates a highpass filter.

Figure 5.12(b): Highpass filter frequency response.

This implies a relationship between their impulse responses:

\[ h_{hp}[n] = e^{j\pi n} h_{lp}[n] = (-1)^n h_{lp}[n] \quad \text{(Eq. 5.33, 5.34)} \]

Here’s a practical example of frequency shifting. Due to the periodicity of the DTFT, shifting a lowpass filter’s frequency response by \(\pi\) radians in the frequency domain transforms it into a highpass filter. This is because \(\omega=0\) are low frequencies, and \(\omega=\pi\) are high frequencies. The frequency shifting property tells us that this spectral shift corresponds to multiplying the lowpass filter’s impulse response by \(e^{j\pi n}\), which is simply \((-1)^n\). This means you can design a highpass filter directly from a lowpass filter by just altering its impulse response in the time domain, which is a powerful shortcut.

5. Conjugation and Conjugate Symmetry

If \(x[n] \stackrel{\mathcal{F}}{\longleftrightarrow} X(e^{j\omega})\):

Conjugation:

Taking the conjugate of a signal in time reflects its spectrum and conjugates it.

\[ x^{*}[n] \stackrel{\mathcal{F}}{\longleftrightarrow} X^{*}\left(e^{-j \omega}\right) \quad \text{(Eq. 5.35)} \]

Conjugate Symmetry (for real \(x[n]\)):

If \(x[n]\) is real, its DTFT exhibits conjugate symmetry.

\[ X\left(e^{j \omega}\right)=X^{*}\left(e^{-j \omega}\right) \quad[x[n] \text { real }] \quad \text{(Eq. 5.36)} \]

This implies:

• \(|X(e^{j\omega})|\) is an even function.
• \(\operatorname{Arg}\{X(e^{j\omega})\}\) is an odd function.
• \(\operatorname{Re}\{X(e^{j\omega})\}\) is an even function.
• \(\operatorname{Im}\{X(e^{j\omega})\}\) is an odd function.

These properties deal with conjugation. Conjugating a signal in the time domain results in a conjugated and frequency-reversed spectrum. Even more importantly, if the time-domain signal \(x[n]\) is real-valued, its DTFT must exhibit conjugate symmetry. This has several important consequences: the magnitude spectrum will be an even function of frequency, and the phase spectrum will be an odd function. Similarly, the real part of the DTFT is even, and the imaginary part is odd. These symmetries are often used to check the validity of a computed DTFT or to deduce properties of the time-domain signal from its spectrum.

6. Differencing and Accumulation

These are the discrete-time counterparts of differentiation and integration.

First Differencing:

The DTFT of the first-difference of a signal.

\[ x[n]-x[n-1] \stackrel{\mathcal{F}}{\longleftrightarrow}\left(1-e^{-j \omega}\right) X\left(e^{j \omega}\right) \quad \text{(Eq. 5.37)} \]

• This is like multiplying by \(j\omega\) in CTFT, but with a discrete-time equivalent.

Accumulation (Running Sum):

The DTFT of the running sum of a signal.

\[ y[n]=\sum_{m=-\infty}^{n} x[m] \stackrel{\mathcal{F}}{\longleftrightarrow} \frac{1}{1-e^{-j \omega}} X\left(e^{j \omega}\right)+\pi X\left(e^{j 0}\right) \sum_{k=-\infty}^{+\infty} \delta(\omega-2 \pi k) \quad \text{(Eq. 5.39)} \]

• The impulse train accounts for the DC (average) value that can arise from summation.

Differencing and accumulation are inverse operations. Taking the first difference of a signal in time corresponds to multiplying its DTFT by \((1-e^{-j\omega})\). This is analogous to multiplying by \(j\omega\) for continuous-time differentiation. Accumulation, or the running sum, is the inverse. Its DTFT involves division by \((1-e^{-j\omega})\), but with an additional impulse term. This impulse term is crucial; it accounts for any DC offset or average value that can build up when summing a signal. For example, summing an impulse results in a step, which has a DC component.

Example 5.8: DTFT of Unit Step \(u[n]\)

Let’s use the accumulation property to find the DTFT of the unit step \(u[n]\).

We know that \(\delta[n] \stackrel{\mathcal{F}}{\longleftrightarrow} 1\). Also, \(u[n]\) is the running sum of \(\delta[n]\): \(u[n]=\sum_{m=-\infty}^{n} \delta[m]\).

Applying the accumulation property (Eq. 5.39) with \(x[n]=\delta[n]\) and \(X(e^{j\omega})=1\):

\[ \mathcal{F}\{u[n]\} = \frac{1}{1-e^{-j \omega}} \cdot 1 + \pi \cdot X(e^{j0}) \sum_{k=-\infty}^{\infty} \delta(\omega-2 \pi k) \]

Since \(X(e^{j\omega})=1\), then \(X(e^{j0})=1\).

Therefore, the DTFT of the unit step is:

\[ U\left(e^{j \omega}\right) = \frac{1}{1-e^{-j \omega}}+\pi \sum_{k=-\infty}^{\infty} \delta(\omega-2 \pi k) \]

This example demonstrates the utility of the accumulation property. We know the DTFT of an impulse is simply 1. Since the unit step is the running sum of an impulse, we can directly apply the accumulation property. The key is to correctly include the impulse train term, which arises because the unit step has a non-zero average value (DC component). Without this term, the DTFT would be incomplete. This result is fundamental and often used.

7. Time Reversal

If \(x[n] \stackrel{\mathcal{F}}{\longleftrightarrow} X(e^{j\omega})\), then reversing the signal in time also reverses its spectrum.

\[ x[-n] \stackrel{\mathcal{F}}{\longleftrightarrow} X\left(e^{-j \omega}\right) \quad \text{(Eq. 5.42)} \]

Derivation:

Let \(y[n]=x[-n]\).

\[ Y\left(e^{j \omega}\right)=\sum_{n=-\infty}^{+\infty} y[n] e^{-j \omega n}=\sum_{n=-\infty}^{+\infty} x[-n] e^{-j \omega n} \quad \text{(Eq. 5.40)} \]

Substitute \(m=-n\):

\[ Y\left(e^{j \omega}\right)=\sum_{m=-\infty}^{+\infty} x[m] e^{-j(-\omega) m}=X\left(e^{-j \omega}\right) \quad \text{(Eq. 5.41)} \]

Time reversal is another straightforward property. If you flip a signal in time, its spectrum also gets flipped (or reversed) in frequency. The derivation is simple and involves a change of variables in the DTFT summation. This property is useful when dealing with signals that have time-reversal symmetry or for analyzing systems that involve time reversal.

8. Time Expansion (Upsampling)

This property is unique in discrete time due to the discrete nature of the time index.

Define \(x_{(k)}[n]\) as:

\[ x_{(k)}[n]= \begin{cases}x[n / k], & \text { if } n \text { is a multiple of } k \\ 0, & \text { if } n \text { is not a multiple of } k .\end{cases} \quad \text{(Eq. 5.44)} \]

• This means inserting \(k-1\) zeros between successive samples of \(x[n]\).
• Intuitively, \(x_{(k)}[n]\) is a “slowed-down” version of \(x[n]\).

If \(x[n] \stackrel{\mathcal{F}}{\longleftrightarrow} X(e^{j\omega})\), then:

\[ x_{(k)}[n] \stackrel{\mathcal{F}}{\longleftrightarrow} X\left(e^{j k \omega}\right) \quad \text{(Eq. 5.45)} \]

• Spreading out the signal in time (upsampling) compresses its spectrum.
• The spectrum \(X(e^{jk\omega})\) becomes periodic with period \(2\pi/k\).

Time expansion, often called upsampling, is distinct in discrete time. We can’t simply scale time by a non-integer factor like in continuous time. Instead, we insert zeros between samples. This effectively “slows down” the signal. The remarkable result is that this time expansion causes the spectrum to be compressed. The original spectrum \(X(e^{j\omega})\) is periodic with \(2\pi\), but \(X(e^{jk\omega})\) will be periodic with \(2\pi/k\). This implies that multiple copies of the baseband spectrum are created within the original \(2\pi\) interval. This property is fundamental to multirate signal processing and digital audio effects.

Visualizing Time Expansion

Figure 5.14: Inverse relationship between time and frequency domains for time expansion.

This figure clearly illustrates the time expansion property. When we insert zeros into the time-domain signal, effectively spreading it out and slowing it down, its Fourier transform becomes compressed. You can see how the original spectrum, which occupies a certain bandwidth, is squeezed into a smaller frequency range, and multiple copies of this compressed spectrum appear within the \(2\pi\) interval. This is a direct visual representation of the inverse relationship between time and frequency domains.

Interactive Demo: Time Expansion (Upsampling)

Observe the effect of upsampling (time expansion) on the spectrum of a simple pulse.

viewof k_upsample = Inputs.range([1, 5], {value: 1, step: 1, label: "Upsampling Factor k"});

This interactive demonstration lets you see the time expansion property in action. As you increase the upsampling factor ‘k’, you’ll observe two things: In the time domain, more zeros are inserted, spreading out the original pulse. In the frequency domain, the spectrum of the signal becomes compressed, and multiple copies of the original spectrum appear within the \(2\pi\) interval. This visual confirmation is crucial for understanding how sampling rate changes affect the frequency content of discrete-time signals.

9. Differentiation in Frequency

This property relates multiplication by \(n\) in the time domain to differentiation in the frequency domain.

\[ n x[n] \stackrel{\mathcal{F}}{\longleftrightarrow} j \frac{d X\left(e^{j \omega}\right)}{d \omega} \quad \text{(Eq. 5.46)} \]

• Derived by differentiating the DTFT analysis equation with respect to \(\omega\).
• Useful for finding the DTFT of signals like ramps (\(n u[n]\)) or for analyzing group delay.

Differentiation in frequency is another powerful property. It states that if you multiply a signal \(x[n]\) by \(n\) in the time domain, its DTFT becomes \(j\) times the derivative of \(X(e^{j\omega})\) with respect to \(\omega\). This is a direct parallel to continuous time. This property is very useful for finding the transforms of signals that involve \(n\) as a multiplier, like a discrete-time ramp function. It also plays a role in understanding concepts like group delay.

10. Parseval’s Relation

Parseval’s relation connects the total energy of a signal in the time domain to its total energy in the frequency domain.

If \(x[n] \stackrel{\mathcal{F}}{\longleftrightarrow} X(e^{j\omega})\), then:

\[ \sum_{n=-\infty}^{+\infty}|x[n]|^{2}=\frac{1}{2 \pi} \int_{2 \pi}\left|X\left(e^{j \omega}\right)\right|^{2} d \omega \quad \text{(Eq. 5.47)} \]

• The left side is the total energy in \(x[n]\).
• The right side is the integral of the energy-density spectrum \(|X(e^{j\omega})|^2 / (2\pi)\) over a \(2\pi\) interval.
• This property is crucial for power calculations and understanding energy distribution in signals.

Parseval’s relation is a fundamental conservation law for signals. It states that the total energy of a signal, calculated by summing the squared magnitudes of its samples in the time domain, is equal to the total energy calculated by integrating the squared magnitude of its DTFT in the frequency domain. The term \(|X(e^{j\omega})|^2\) is often called the energy-density spectrum, indicating how energy is distributed across different frequencies. This property is vital for analyzing signal power, noise, and overall system performance.

Example 5.10: Analyzing Signal Properties from DTFT

Given the DTFT \(X(e^{j\omega})\) for \(-\pi \leq \omega \leq \pi\) as shown:

Figure 5.16: Magnitude and phase of \(X(e^{j\omega})\).

Let’s determine if \(x[n]\) is: periodic, real, even, and/or of finite energy.

1. Periodic? No. Its DTFT is not an impulse train.
2. Real? Yes. \(|X(e^{j\omega})|\) is even, and \(\operatorname{Arg}\{X(e^{j\omega})\}\) is odd. (Conjugate Symmetry)
3. Even? No. If \(x[n]\) were even and real, \(X(e^{j\omega})\) would be real and even. Here, \(X(e^{j\omega}) = |X(e^{j\omega})|e^{-j2\omega}\), which is not real (due to the phase term).
4. Finite Energy? Yes. The integral of \(|X(e^{j\omega})|^2\) over \(-\pi \leq \omega \leq \pi\) will be a finite value, as seen from the plot. (Parseval’s Relation)

This example demonstrates how we can use the DTFT properties to infer characteristics of the time-domain signal without actually computing the inverse DTFT. By examining the given magnitude and phase plots of \(X(e^{j\omega})\), we can systematically apply the properties:

1. Periodicity in time would mean an impulse train in frequency; this is not the case.
2. For a real signal, magnitude is even, phase is odd; this holds true.
3. For a real and even signal, the DTFT must be real and even. Here, the phase is non-zero, making the DTFT complex, so it’s not even.
4. The magnitude squared is clearly integrable and finite over \(2\pi\), so the signal has finite energy according to Parseval’s relation. This kind of analysis is very practical in signal processing.

Advanced Properties (Brief Mention)

We will explore these in more detail in upcoming sections:

• Convolution Property:

  \(x[n] * h[n] \stackrel{\mathcal{F}}{\longleftrightarrow} X(e^{j\omega}) H(e^{j\omega})\)

  (Convolution in time becomes multiplication in frequency)

• Multiplication Property:

  \(x[n] y[n] \stackrel{\mathcal{F}}{\longleftrightarrow} \frac{1}{2\pi} \int_{2\pi} X(e^{j\theta}) Y(e^{j(\omega-\theta)}) d\theta\)

  (Multiplication in time becomes periodic convolution in frequency)

• Duality:

  A powerful concept relating properties between time and frequency domains, and even between continuous-time and discrete-time domains.

Finally, we briefly mention two more critical properties that we’ll cover in depth later: convolution and multiplication. These are arguably the most important properties for system analysis. Convolution in the time domain, which describes the output of an LTI system, becomes simple multiplication in the frequency domain. Conversely, multiplication in the time domain, such as in amplitude modulation, corresponds to periodic convolution in the frequency domain. We’ll also explore duality, a concept that highlights the symmetric nature of the Fourier transform between the time and frequency domains, and even between continuous and discrete domains.