Table of Contents

• Chapter 4
• Chapter 5
• Chapter 6

Chapter 4

Section 4.3: Properties of the Continuous-Time Fourier Transform

1. If you add two signals in the time domain, how do you find the Fourier transform of the sum?



2. When a signal x(t) is delayed in time, how does its Fourier transform change?



3. Does a real signal have a real Fourier transform?

Section 4.4: The Convolution Property

1. If two signals are convolved in the time domain, what is the corresponding operation in the frequency domains? In other words if y(t)=h(t)*x(t), how is the Fourier transform of y(t) related to the Fourier transforms of x(t) and h(t)?



2. How is the impulse response of an LTI system related to its frequency response?

Section 4.5: The Multiplication Property

1. If two signals are multiplied in the time domain, what is the corresponding operation in the frequency domain? In other words if r(t)=p(t)s(t), how can you find the Fourier transform of r(t) from the Fourier transforms of p(t) and s(t)?

Section 4.7: Systems Characterized By Linear Constant-Coefficient Differential Equations

1. Given a differential equation description of an LTI system, is it easy to obtain its frequency response?

Chapter 5

Section 5.1: Representation of Aperiodic Signals: The Discrete-Time Fourier Transform

1. What do the analysis and synthesis equations for the DT Fourier transform look like?



2. What are the major differences between the continuous-time and discrete-time Fourier transforms?



3. How can we relate the FS coefficients of a periodic signal x[n] to the DTFT of an aperiodic signal that is equal to x[n] over 1 period and zero otherwise?

Section 5.2: The Fourier Transform for Periodic Signals

1. Give the DTFS coefficients of a periodic signal can we obtain an expression for its Fourier transform? How?



2. What does the DTFT of a periodic signal look like?

Section 5.3: Properties of the Discrete-Time Fourier Transform

1. When is the DTFT not periodic?



2. How can you prove that the DTFT satisfies the linearity property?



3. A shift in the time-domain corresponds to multiplication by a complex exponential in the frequency-domain. Is there an analogous property for a shift in the frequency-domain?



4. Can you define the signal x[an] for any value of a?



5. What does Parseval's relation tell you?

Section 5.4: The Convolution Property

1. If two DT LTI systems are cascaded, how can you find the frequency response of the overall system? What is the impulse response of the overall system?



2. Does every DT LTI system have a frequency response? If not, what condition guarantees that the frequency response exists?

Section 5.5: The Multiplication Property

1. Multiplication of two continuous-time signals corresponds to convolution of their Fourier transforms? Is the same true for discrete-time signals?



2. What is periodic convolution?

Section 5.8: Systems Characterized By Linear Constant-Coefficient Difference Equations

1. Given a difference equation, how can you find the system's frequency response?



2. Given a frequency response that is a ratio of polynomials, can you find a difference equation to describe the system? How?

Chapter 6

Section 6.1: The Magnitude-Phase Representation of the Fourier Transform

1. We can think of the Fourier transform as a "sum" of complex exponentials at different frequencies. In that case:
1. What information does the magnitude of the Fourier transform provide?
2. What information does the phase of the Fourier transform provide?

Section 6.2: The Magnitude-Phase Representation of the Frequency Response of LTI Systems

1. What effect does a linear phase system have on input signals?



2. What effect does a nonlinear phase system have on input signals?



3. What is dispersion?



4. What is an allpass system?



5. What is a Bode plot?

Section 6.3: Time-Domain Properties of Ideal Frequency-Selective Filters

1. As the bandwidth of an ideal lowpass filter gets wider, what happens to its impulse response?

Section 6.4: Time-Domain and Frequency-Domain Aspects of Nonideal Filters

1. Why are ideal filters not always desirable in practice?



2. What are some of the time/frequency domain tradeoff involved in using nonideal filters?

Section 6.5: First-Order and Second-Order CT Systems

1. What is ringing?



2. What is overshoot?



3. How is a Bode plot constructed?

Section 6.6: First-Order and Second-Order DT Systems

1. What is the difference equation for a 1st-order DT system?
   
   
2. What determines the rate at which a 1st-order DT system responds?

Section 6.7: Examples of Time- and Frequency-Domain Analysis of Systems

1. What is a non-recursive system?