1-Can we calculate the Fourier Transform for a function represented by Fourier Series? Elaborate. 2-What happens...
1-Can we calculate the Fourier Transform for a function represented by Fourier Series? Elaborate.
2-What happens if we sample with a frequency that is less than half the maximum frequency of the sampled signal?
3-Describe in your words what is Fourier Series and its relation to periodic signals. Mention whether it is a time domain or frequency domain representation
Homework Answers
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1-Can we calculate the Fourier Transform for a function
represented by Fourier Series? Elaborate.
2-What happens...
5.1-7 Consider a bandlimited signal g1(C) whose Fourier transform is (a) If g1(t) is uniformly sa...
5.1-7 Consider a bandlimited signal g1(C) whose Fourier transform is (a) If g1(t) is uniformly sampled at the rate of fs400 Hz, show the resulting spectrum of the ideally sampled signal. (b) If we attempt to reconstruct gi (t) from the samples in Part (a), what will be the recovered analog signal in both time and frequency domains? (c) Determine another analog signal G2(f) in frequency domain such that its samples at = 400 Hz will lead to the same...
(b) Let X(ju) denote the Fourier transform of the signal r(t) shown in the figure x(t) 2 -2 1 2 Using the properties of the Fourier transform (and without explicitly evaluating X(jw)), ii. (5 pts)...
(b) Let X(ju) denote the Fourier transform of the signal r(t) shown in the figure x(t) 2 -2 1 2 Using the properties of the Fourier transform (and without explicitly evaluating X(jw)), ii. (5 pts) Find2X(jw)dw. Hint: Apply the definition of the inverse Fourier transform formula, and you can also recall the time shift property for Fourier Transform. (c) (5 pts) Fourier Series. Consider the periodic signal r(t) below: 1 x(t) 1 -2 ·1/4 Transform r(t) into its Fourier Series...
Q. 2 A continuous time signal x(t) has the Continuous Time Fourier Transform shown in Fig...
Q. 2 A continuous time signal x(t) has the Continuous Time Fourier Transform shown in Fig 2. Xc() -80007 0 80001 2 (rad/s) Fig 2 According to the sampling theorem, find the maximum allowable sampling period T for this signal. Also plot the Fourier Transforms of the sampled signal X:(j) and X(elo). Label the resulting signals appropriately (both in frequency and amplitude axis). Assuming that the sampling period is increased 1.2 times, what is the new sampling frequency 2? What...
1. Consider the complex-valued signal r(t) with Fourier transform as shown in the figure. Keep in...
1. Consider the complex-valued signal r(t) with Fourier transform as shown in the figure. Keep in mind that there are no symmetry properties this signal satisfies in the fre- quency domain. In particular, the Fourier transform is zero for negative frequencies Suppose we impulse-train sample x(t) at the rate of 500 samples/second. 200 400 600 800 1000 FREQUENCY (Hz.) (a) Sketch the Fourier transform of the impulse-train sampled signal in the range of frequencies from -1000 Hz. to 1000 Hz....
10ρ 18ρ A signal (t) has the Fourier transform X(jw) indicated in the figure. The signal is sampled to obtain the discrete time signal 1. Sketch the Fourier transform Xr(jw) of x[n] for T-to. 2. Can...
10ρ 18ρ A signal (t) has the Fourier transform X(jw) indicated in the figure. The signal is sampled to obtain the discrete time signal 1. Sketch the Fourier transform Xr(jw) of x[n] for T-to. 2. Can x(t) be recovered for T? How? What is the maximum value of T so that r(t) can be recovered?
10ρ 18ρ A signal (t) has the Fourier transform X(jw) indicated in the figure. The signal is sampled to obtain the discrete time signal 1....
10. Find the Fourier transform of a continuous-time signal x(t) = 10e Su(t). Plot the magnitude...
10. Find the Fourier transform of a continuous-time signal x(t) = 10e Su(t). Plot the magnitude spectrum and the phase spectrum. If the signal is going to be sampled, what should be the minimum sampling frequency so that the aliasing error is less than 0.1 % of the maximum original magnitude at half the sampling frequency. 11. A signal x(t) = 5cos(2nt + 1/6) is sampled at every 0.2 seconds. Find the sequence obtained over the interval 0 st 3...
Points) Consider the signal s(t) with Fourier Transform 10 1+ω. S(a) figure below, we impulse sam...
points) Consider the signal s(t) with Fourier Transform 10 1+ω. S(a) figure below, we impulse sample s) at a frequency o, rads/second, e signal sa(t). Can you find a finite sampling frequency o such that ly recover s(t) from so()? If so, find it. If not, explain why not. a) (5 pts) In ting in the can perfectly you s (t) sa(t) →| Impulse sample at- rate o b) (5 pts) Now suppose we filter the signal s() with an...
Problem (3) a) A periodic square wave signal x(t) is shown below, it is required to answer the below questions: x(t) 1. What is the period and the duration of such a signal? 2. Determine the fund...
Problem (3) a) A periodic square wave signal x(t) is shown below, it is required to answer the below questions: x(t) 1. What is the period and the duration of such a signal? 2. Determine the fundamental frequency. 3. Calculate the Trigonometric Fourier Series and sketch the amplitude spectrum and phase spectrum of the signal x(t) for the first 5 harmonics. b) Find the Continuous Time Fourier Series (CTFS) and Continuous Time Fourier Transform (CTFT) of the following periodic signals...
Don't need to do #1. Please go into detail on how you solved #2 and #3 The Fourier transform of the signal r(t) is given by the following figure (X(jw)0 for w> 20) X(ju) 0.8 0.6 0.4 0.2 -10 10...
Don't need to do #1. Please go into detail on how you solved #2
and #3
The Fourier transform of the signal r(t) is given by the following figure (X(jw)0 for w> 20) X(ju) 0.8 0.6 0.4 0.2 -10 10 20 m Page 4 of 5 Final S09 EE315 Signals & Systems The signal is sampled to obtain the signal withFourier transform Xlw 1. (5p) What is the minimum sampling frequency w 2. (10p) Now suppose that the sampling frequency...
Suppose that r(l) is a band-limited signal with the bandwidth W. Suppose that we sampled this...
Suppose that r(l) is a band-limited signal with the bandwidth W. Suppose that we sampled this signal with the sainpling interval T, to generate the sample sequence 1 TLI suppose that 2n/T is larger than the Nyquist rate 2W Given rn, we reconstructed a conius time signal ( using the zero-order-hold method. In other words, rr(l) n for L E [nT, (n +1)T;). In the last lecture, we derived that where s(), as usual, denotes the continuous time representation of...
5.1-7 Consider a bandlimited signal g1(C) whose Fourier transform is (a) If g1(t) is uniformly sampled at the rate of fs400 Hz, show the resulting spectrum of the ideally sampled signal. (b) If we attempt to reconstruct gi (t) from the samples in Part (a), what will be the recovered analog signal in both time and frequency domains? (c) Determine another analog signal G2(f) in frequency domain such that its samples at = 400 Hz will lead to the same...
(b) Let X(ju) denote the Fourier transform of the signal r(t) shown in the figure x(t) 2 -2 1 2 Using the properties of the Fourier transform (and without explicitly evaluating X(jw)), ii. (5 pts) Find2X(jw)dw. Hint: Apply the definition of the inverse Fourier transform formula, and you can also recall the time shift property for Fourier Transform. (c) (5 pts) Fourier Series. Consider the periodic signal r(t) below: 1 x(t) 1 -2 ·1/4 Transform r(t) into its Fourier Series...
Q. 2 A continuous time signal x(t) has the Continuous Time Fourier Transform shown in Fig 2. Xc() -80007 0 80001 2 (rad/s) Fig 2 According to the sampling theorem, find the maximum allowable sampling period T for this signal. Also plot the Fourier Transforms of the sampled signal X:(j) and X(elo). Label the resulting signals appropriately (both in frequency and amplitude axis). Assuming that the sampling period is increased 1.2 times, what is the new sampling frequency 2? What...
1. Consider the complex-valued signal r(t) with Fourier transform as shown in the figure. Keep in mind that there are no symmetry properties this signal satisfies in the fre- quency domain. In particular, the Fourier transform is zero for negative frequencies Suppose we impulse-train sample x(t) at the rate of 500 samples/second. 200 400 600 800 1000 FREQUENCY (Hz.) (a) Sketch the Fourier transform of the impulse-train sampled signal in the range of frequencies from -1000 Hz. to 1000 Hz....
10ρ 18ρ A signal (t) has the Fourier transform X(jw) indicated in the figure. The signal is sampled to obtain the discrete time signal 1. Sketch the Fourier transform Xr(jw) of x[n] for T-to. 2. Can x(t) be recovered for T? How? What is the maximum value of T so that r(t) can be recovered?
10ρ 18ρ A signal (t) has the Fourier transform X(jw) indicated in the figure. The signal is sampled to obtain the discrete time signal 1....
10. Find the Fourier transform of a continuous-time signal x(t) = 10e Su(t). Plot the magnitude spectrum and the phase spectrum. If the signal is going to be sampled, what should be the minimum sampling frequency so that the aliasing error is less than 0.1 % of the maximum original magnitude at half the sampling frequency. 11. A signal x(t) = 5cos(2nt + 1/6) is sampled at every 0.2 seconds. Find the sequence obtained over the interval 0 st 3...
points) Consider the signal s(t) with Fourier Transform 10 1+ω. S(a) figure below, we impulse sample s) at a frequency o, rads/second, e signal sa(t). Can you find a finite sampling frequency o such that ly recover s(t) from so()? If so, find it. If not, explain why not. a) (5 pts) In ting in the can perfectly you s (t) sa(t) →| Impulse sample at- rate o b) (5 pts) Now suppose we filter the signal s() with an...
Problem (3) a) A periodic square wave signal x(t) is shown below, it is required to answer the below questions: x(t) 1. What is the period and the duration of such a signal? 2. Determine the fundamental frequency. 3. Calculate the Trigonometric Fourier Series and sketch the amplitude spectrum and phase spectrum of the signal x(t) for the first 5 harmonics. b) Find the Continuous Time Fourier Series (CTFS) and Continuous Time Fourier Transform (CTFT) of the following periodic signals...
Don't need to do #1. Please go into detail on how you solved #2
and #3
The Fourier transform of the signal r(t) is given by the following figure (X(jw)0 for w> 20) X(ju) 0.8 0.6 0.4 0.2 -10 10 20 m Page 4 of 5 Final S09 EE315 Signals & Systems The signal is sampled to obtain the signal withFourier transform Xlw 1. (5p) What is the minimum sampling frequency w 2. (10p) Now suppose that the sampling frequency...
Suppose that r(l) is a band-limited signal with the bandwidth W. Suppose that we sampled this signal with the sainpling interval T, to generate the sample sequence 1 TLI suppose that 2n/T is larger than the Nyquist rate 2W Given rn, we reconstructed a conius time signal ( using the zero-order-hold method. In other words, rr(l) n for L E [nT, (n +1)T;). In the last lecture, we derived that where s(), as usual, denotes the continuous time representation of...
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