5.2 Lab 4: lab

Implementation

You will use the FFT to compute the spectrum of a windowed input. For your implementation, use a 64-point Hammingwindow. You may use the MATLAB function save_coef (available as save_coef.m to save the window to a file that you can then include in your code with the .copy directive.

Fft usage

The FFT routine fft.asm computes an in-place, complex FFT. The length of the FFT is defined as a label K_FFT_SIZE , and the algorithm assumes that the input starts at data memory location fft_data . To have your code assemble for a 64-point FFT, you will haveto include the following label definitions in your code.

In addition to defining these constants, you will have to include twiddle-factor tables for the FFT. Copy these tables ( twiddle1 and twiddle2 ) into your work directory. Note that the tables are each 512 points long, representing valuesfrom 0 to just shy of 180 degrees, and must be accessed as a circular buffer. To include these tables at the properlocation in memory with the appropriate labels referenced by the FFT, use the following:

To run the FFT code, use the instruction call fft where fft is a label at the beginning of the available fft.asm code.

The FFT provided requires that the input be in bit-reversed order, with alternating real and imaginary components. Bit-reversed addressing is aconvenient way to order input $x(n)$ into a FFT so that the output $X(k)$ is in sequential order ( i.e. , $X(0)$ , $X(1)$ ,, $X(N-1)$ for an $N$ -point FFT). The table below illustrates the bit-reversed order for an eight-point sequence.

The following routine performs the bit-reversed reordering of the input data. The routine assumes that the input isstored in data memory starting at the location labeled input_data and consists of alternating real and imaginary parts. Because our input data in this exercise ispurely real, you will have to set the imaginary parts to zero by zeroing out every other memory location.

As mentioned, in the above code input_data is a label indicating the start of the input data and fft_data is a label indicating the start of a circular buffer where the bit-reversed data will be written.Include the bit_rev routine in your code and call it using the call bit_rev command in the appropriate location. Note that although AR7 is not used by the bit-reversed routine directly, it is used extensively in the FFT routine to keep track of the start ofthe FFT data space.

In general, to have a pointer index memory in bit-reversed order, the AR0 register needs to be set to one-half the length of the circular buffer; a statement suchas ARx+0B is used to move the ARx pointer to the next location. For more information regarding the bit-reversed addressing mode, refer to page 5-18 in the CPU and Peripherals manual. See Figure 4-10 in the Applications Guide to view the ordering of the data expected by the FFT routine.

Displaying the spectrum

Once the DFT has been computed, calculate the squared-magnitude of the spectrum for display.

Because the squared magnitude is always nonnegative, you can replace one of the magnitude values with a -1.0 as a triggerpulse for display on the oscilloscope. This is easily performed by replacing the DC term, $k=0$ , with a -1.0 when copying the magnitude values to the output buffer. The trigger pulse is necessary for theoscilloscope to lock to a specific point in the spectrum and keep the spectrum fixed on the scope.